diff --git a/src/Utilities/model.jl b/src/Utilities/model.jl
index 0aa4690640..6586b95e42 100644
--- a/src/Utilities/model.jl
+++ b/src/Utilities/model.jl
@@ -312,9 +312,16 @@ end
 
 function MOI.add_constraint(
     model::AbstractModel,
-    func::MOI.AbstractFunction,
-    set::MOI.AbstractSet,
-)
+    func::F,
+    set::S,
+) where {F<:MOI.AbstractFunction,S<:MOI.AbstractSet}
+    # We check supports_constraint here because it is a common practice for
+    # AbstractModels to declare that they do not support particular constraints,
+    # even though the underlying `.constraints` object does. See, for example,
+    # the various models in MOI.FileFormats.
+    if !MOI.supports_constraint(model, F, S)
+        throw(MOI.UnsupportedConstraint{F,S}())
+    end
     return MOI.add_constraint(model.constraints, func, set)
 end
 
@@ -328,10 +335,17 @@ end
 
 function MOI.add_constraint(
     model::AbstractModel,
-    f::MOI.VariableIndex,
-    s::MOI.AbstractScalarSet,
-)
-    return MOI.add_constraint(model.variables, f, s)
+    func::MOI.VariableIndex,
+    set::S,
+) where {S<:MOI.AbstractScalarSet}
+    # We check supports_constraint here because it is a common practice for
+    # AbstractModels to declare that they do not support particular constraints,
+    # even though the underlying `.constraints` object does. See, for example,
+    # the various models in MOI.FileFormats.
+    if !MOI.supports_constraint(model, MOI.VariableIndex, S)
+        throw(MOI.UnsupportedConstraint{MOI.VariableIndex,S}())
+    end
+    return MOI.add_constraint(model.variables, func, set)
 end
 
 # MOI.NumberOfConstraints
diff --git a/test/Utilities/model.jl b/test/Utilities/model.jl
index 0e9817a139..ba52d51f95 100644
--- a/test/Utilities/model.jl
+++ b/test/Utilities/model.jl
@@ -515,6 +515,35 @@ function test_vector_of_constraints_list_of_constraint_attributes_set()
     return
 end
 
+function test_copy_to_unsupported_scalar_variable()
+    F, S = MOI.VariableIndex, MOI.GreaterThan{Float64}
+    model = MOI.Utilities.Model{Float64}()
+    x, _ = MOI.add_constrained_variable(model, MOI.GreaterThan(1.0))
+    dest = MOI.FileFormats.CBF.Model()
+    @test_throws(MOI.UnsupportedConstraint{F,S}, MOI.copy_to(dest, model))
+    return
+end
+
+function test_copy_to_unsupported_vector_variables()
+    F, S = MOI.VectorOfVariables, MOI.AllDifferent
+    model = MOI.Utilities.Model{Float64}()
+    x, _ = MOI.add_constrained_variables(model, MOI.AllDifferent(3))
+    dest = MOI.FileFormats.CBF.Model()
+    @test_throws(MOI.UnsupportedConstraint{F,S}, MOI.copy_to(dest, model))
+    return
+end
+
+function test_copy_to_unsupported_scalar_function()
+    F, S = MOI.ScalarNonlinearFunction, MOI.EqualTo{Float64}
+    model = MOI.Utilities.Model{Float64}()
+    x = MOI.add_variable(model)
+    f = MOI.ScalarNonlinearFunction(:log, Any[x])
+    MOI.add_constraint(model, f, MOI.EqualTo(0.0))
+    dest = MOI.FileFormats.CBF.Model()
+    @test_throws(MOI.UnsupportedConstraint{F,S}, MOI.copy_to(dest, model))
+    return
+end
+
 end  # module
 
 TestModel.runtests()