diff --git a/src/FileFormats/NL/NL.jl b/src/FileFormats/NL/NL.jl
index 5029e49776..90952e5903 100644
--- a/src/FileFormats/NL/NL.jl
+++ b/src/FileFormats/NL/NL.jl
@@ -880,6 +880,14 @@ function MOI.get(model::Model, attr::MOI.AbstractModelAttribute)
     return MOI.get(inner, attr)
 end
 
+# It doesn't need the inner model
+function MOI.get(
+    model::Model,
+    attr::Union{MOI.VariableBridgingCost,MOI.ConstraintBridgingCost},
+)
+    return MOI.get_fallback(model, attr)
+end
+
 function MOI.get(
     model::Model,
     attr::MOI.AbstractConstraintAttribute,
diff --git a/src/Test/test_attribute.jl b/src/Test/test_attribute.jl
index a8fa9faaac..a5bb796f27 100644
--- a/src/Test/test_attribute.jl
+++ b/src/Test/test_attribute.jl
@@ -393,3 +393,55 @@ function test_attribute_unsupported_constraint(model::MOI.ModelLike, ::Config)
 end
 
 version_added(::typeof(test_attribute_unsupported_constraint)) = v"1.9.0"
+
+"""
+    test_attribute_VariableBridgingCost(model::MOI.ModelLike, config::Config)
+
+Test that, for every set `S` that the model claims to support via
+`supports_add_constrained_variable(s)`, the corresponding
+[`MOI.VariableBridgingCost`](@ref) attribute returns a finite value.
+
+This is the variable-side analog of the `ConstraintBridgingCost` check in
+`_basic_constraint_test_helper`.
+
+The fallback works for most model but it may need custom method for some MOI
+layers (see https://github.com/jump-dev/MathOptInterface.jl/pull/3001#issuecomment-4468198935).
+
+This test is here to catch that.
+"""
+function test_attribute_VariableBridgingCost(
+    model::MOI.ModelLike,
+    ::Config{T},
+) where {T}
+    for S in Any[
+        MOI.GreaterThan{T},
+        MOI.LessThan{T},
+        MOI.EqualTo{T},
+        MOI.Interval{T},
+        MOI.Integer,
+        MOI.ZeroOne,
+        MOI.Semicontinuous{T},
+        MOI.Semiinteger{T},
+    ]
+        if MOI.supports_add_constrained_variable(model, S)
+            @test MOI.get(model, MOI.VariableBridgingCost{S}()) < Inf
+        end
+    end
+    for S in Any[
+        MOI.Reals,
+        MOI.Zeros,
+        MOI.Nonnegatives,
+        MOI.Nonpositives,
+        MOI.SecondOrderCone,
+        MOI.RotatedSecondOrderCone,
+        MOI.ExponentialCone,
+        MOI.PositiveSemidefiniteConeTriangle,
+    ]
+        if MOI.supports_add_constrained_variables(model, S)
+            @test MOI.get(model, MOI.VariableBridgingCost{S}()) < Inf
+        end
+    end
+    return
+end
+
+version_added(::typeof(test_attribute_VariableBridgingCost)) = v"1.52.0"
diff --git a/src/Test/test_basic_constraint.jl b/src/Test/test_basic_constraint.jl
index f9d0030d66..4bb529ac6a 100644
--- a/src/Test/test_basic_constraint.jl
+++ b/src/Test/test_basic_constraint.jl
@@ -255,6 +255,16 @@ function _basic_constraint_test_helper(
     ###
     @requires MOI.supports_constraint(model, F, S)
     ###
+    ### Test MOI.ConstraintBridgingCost
+    ###
+    # If `supports_constraint(F, S)` returns `true`, then the model must be
+    # able to handle that pair (possibly via bridging), so the bridging cost
+    # must be finite. The fallback works for most model but it may need
+    # custom method for some MOI layer (see
+    # https://github.com/jump-dev/MathOptInterface.jl/pull/3001#issuecomment-4468198935)
+    # This test is here to catch that.
+    @test MOI.get(model, MOI.ConstraintBridgingCost{F,S}()) < Inf
+    ###
     ### Test MOI.NumberOfConstraints
     ###
     @test MOI.get(model, MOI.NumberOfConstraints{F,S}()) == 0
diff --git a/src/Utilities/cachingoptimizer.jl b/src/Utilities/cachingoptimizer.jl
index f3392bbf07..0df5e5863f 100644
--- a/src/Utilities/cachingoptimizer.jl
+++ b/src/Utilities/cachingoptimizer.jl
@@ -897,6 +897,16 @@ function MOI.get(model::CachingOptimizer, attr::MOI.AbstractModelAttribute)
     return _get_model_attribute(model, attr)
 end
 
+function MOI.get(
+    model::CachingOptimizer,
+    attr::Union{MOI.VariableBridgingCost,MOI.ConstraintBridgingCost},
+)::Float64
+    if state(model) == NO_OPTIMIZER
+        return MOI.get(model.model_cache, attr)
+    end
+    return MOI.get(model.optimizer, attr)
+end
+
 function MOI.get(
     model::CachingOptimizer,
     attr::MOI.TerminationStatus,
diff --git a/src/Utilities/universalfallback.jl b/src/Utilities/universalfallback.jl
index 8a6f9aa063..b3a57b7483 100644
--- a/src/Utilities/universalfallback.jl
+++ b/src/Utilities/universalfallback.jl
@@ -372,6 +372,36 @@ function MOI.get(
     return _get(uf, attr)
 end
 
+function MOI.get(
+    uf::UniversalFallback,
+    attr::MOI.ConstraintBridgingCost{F,S},
+) where {F,S}
+    if MOI.supports_constraint(uf.model, F, S)
+        return MOI.get(uf.model, attr)
+    end
+    return 0.0
+end
+
+function MOI.get(
+    uf::UniversalFallback,
+    attr::MOI.VariableBridgingCost{S},
+) where {S<:MOI.AbstractScalarSet}
+    if MOI.supports_add_constrained_variable(uf.model, S)
+        return MOI.get(uf.model, attr)
+    end
+    return 0.0
+end
+
+function MOI.get(
+    uf::UniversalFallback,
+    attr::MOI.VariableBridgingCost{S},
+) where {S<:MOI.AbstractVectorSet}
+    if MOI.supports_add_constrained_variables(uf.model, S)
+        return MOI.get(uf.model, attr)
+    end
+    return 0.0
+end
+
 function MOI.get(
     uf::UniversalFallback,
     attr::MOI.AbstractConstraintAttribute,
diff --git a/test/Bridges/Constraint/test_IntervalToHyperRectangleBridge.jl b/test/Bridges/Constraint/test_IntervalToHyperRectangleBridge.jl
index 63a992bc62..4b1e0b8c38 100644
--- a/test/Bridges/Constraint/test_IntervalToHyperRectangleBridge.jl
+++ b/test/Bridges/Constraint/test_IntervalToHyperRectangleBridge.jl
@@ -38,7 +38,7 @@ function test_basic(T)
         MOI.Utilities.UniversalFallback(MOI.Utilities.Model{T}()),
     )
     bridged_mock = MOI.Bridges.Constraint.IntervalToHyperRectangle{T}(mock)
-    config = MOI.Test.Config()
+    config = MOI.Test.Config(T)
     MOI.Test.runtests(
         bridged_mock,
         config,
diff --git a/test/FileFormats/NL/test_NL.jl b/test/FileFormats/NL/test_NL.jl
index 88c04f8aea..f62916cc79 100644
--- a/test/FileFormats/NL/test_NL.jl
+++ b/test/FileFormats/NL/test_NL.jl
@@ -1542,6 +1542,17 @@ function test_unsupported_kwarg()
     return
 end
 
+function test_VariableBridgingCost()
+    model = NL.Model()
+    attr = MOI.VariableBridgingCost{MOI.LessThan{Float64}}()
+    @test MOI.get(model, attr) == 0
+    attr = MOI.ConstraintBridgingCost{MOI.VariableIndex,MOI.LessThan{Float64}}()
+    @test MOI.get(model, attr) == 0
+    attr = MOI.VariableBridgingCost{MOI.SecondOrderCone}()
+    @test isinf(MOI.get(model, attr))
+    return
+end
+
 end
 
 TestNLModel.runtests()
diff --git a/test/Utilities/test_cachingoptimizer.jl b/test/Utilities/test_cachingoptimizer.jl
index 9d7b38abc1..3b26306a0e 100644
--- a/test/Utilities/test_cachingoptimizer.jl
+++ b/test/Utilities/test_cachingoptimizer.jl
@@ -1465,6 +1465,24 @@ function test_rethrow_set_model_attribute()
     return
 end
 
+function test_BridgingCost_NO_OPTIMIZER()
+    cache = MOI.Utilities.Model{Float64}()
+    model = MOI.Utilities.CachingOptimizer(cache, MOI.Utilities.AUTOMATIC)
+    @test MOI.Utilities.state(model) == MOI.Utilities.NO_OPTIMIZER
+    @test MOI.get(model, MOI.VariableBridgingCost{MOI.LessThan{Float64}}()) ==
+          0.0
+    @test MOI.get(model, MOI.VariableBridgingCost{MOI.LessThan{Int}}()) == Inf
+    @test MOI.get(
+        model,
+        MOI.ConstraintBridgingCost{MOI.VariableIndex,MOI.LessThan{Float64}}(),
+    ) == 0.0
+    @test MOI.get(
+        model,
+        MOI.ConstraintBridgingCost{MOI.VariableIndex,MOI.LessThan{Int}}(),
+    ) == Inf
+    return
+end
+
 end  # module
 
 TestCachingOptimizer.runtests()
diff --git a/test/Utilities/test_universalfallback.jl b/test/Utilities/test_universalfallback.jl
index d845e270a1..da5251f642 100644
--- a/test/Utilities/test_universalfallback.jl
+++ b/test/Utilities/test_universalfallback.jl
@@ -530,6 +530,48 @@ function test_set_inner_constraint_attribute()
     return
 end
 
+function test_bridging_cost_consistent_with_supports()
+    # `UniversalFallback.supports_constraint` and
+    # `supports_add_constrained_variable(s)` accept absolutely anything by
+    # forwarding through their `is_bridged`-style catch-all. The bridging-cost
+    # attributes must agree: they should never return `Inf` for a pair that
+    # `supports_*` claims to support, because that would make
+    # `LazyBridgeOptimizer` treat the node as unreachable and break graph
+    # construction. Use `Model{BigFloat}` so the inner model genuinely does
+    # not support `*Cone{Float64}`, exercising the case where
+    # `UniversalFallback` extends support beyond the inner.
+    model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{BigFloat}())
+    for (F, S) in (
+        # inner supports natively
+        (MOI.ScalarAffineFunction{BigFloat}, MOI.LessThan{BigFloat}),
+        (MOI.VectorOfVariables, MOI.PowerCone{BigFloat}),
+        # inner does not support; UF stores in its own dict
+        (MOI.ScalarAffineFunction{Float64}, MOI.LessThan{Float64}),
+        (MOI.VectorOfVariables, MOI.PowerCone{Float64}),
+        (MOI.VectorAffineFunction{BigFloat}, MOI.Test.UnknownVectorSet),
+    )
+        @test MOI.supports_constraint(model, F, S)
+        @test MOI.get(model, MOI.ConstraintBridgingCost{F,S}()) < Inf
+    end
+    for S in (
+        MOI.GreaterThan{BigFloat},
+        MOI.Integer,
+        MOI.GreaterThan{Float64},  # not natively supported by Model{BigFloat}
+    )
+        @test MOI.supports_add_constrained_variable(model, S)
+        @test MOI.get(model, MOI.VariableBridgingCost{S}()) < Inf
+    end
+    for S in (
+        MOI.Nonnegatives,
+        MOI.PowerCone{BigFloat},
+        MOI.PowerCone{Float64},  # not natively supported by Model{BigFloat}
+    )
+        @test MOI.supports_add_constrained_variables(model, S)
+        @test MOI.get(model, MOI.VariableBridgingCost{S}()) < Inf
+    end
+    return
+end
+
 end  # module
 
 TestUniversalFallback.runtests()