diff --git a/src/constraints.jl b/src/constraints.jl
index 9b49a9903a6..7564defba18 100644
--- a/src/constraints.jl
+++ b/src/constraints.jl
@@ -134,8 +134,19 @@ end
 
 Get a constraint's name attribute.
 """
-function name(con_ref::ConstraintRef{<:AbstractModel,<:_MOICON})
-    return MOI.get(con_ref.model, MOI.ConstraintName(), con_ref)::String
+function name(con_ref::ConstraintRef{<:AbstractModel,C}) where {C<:_MOICON}
+    model = owner_model(con_ref)
+    if !MOI.supports(backend(model), MOI.ConstraintName(), C)
+        return ""
+    end
+    return MOI.get(model, MOI.ConstraintName(), con_ref)::String
+end
+
+# The name of VariableIndex constraints is empty.
+function name(
+    ::ConstraintRef{<:AbstractModel,<:MOI.ConstraintIndex{MOI.VariableIndex}},
+)
+    return ""
 end
 
 """
@@ -554,8 +565,10 @@ function add_constraint(
         model.shapes[cindex] = cshape
     end
     con_ref = ConstraintRef(model, cindex, cshape)
-    if !(func isa MOI.VariableIndex) && !isempty(name)
-        # Don't set names for VariableIndex constraints!
+    # Only set names if appropriate!
+    if !(func isa MOI.VariableIndex) &&
+       !isempty(name) &&
+       MOI.supports(backend(model), MOI.ConstraintName(), typeof(cindex))
         set_name(con_ref, name)
     end
     return con_ref
diff --git a/src/variables.jl b/src/variables.jl
index 1a3959fd6cb..daf349691b4 100644
--- a/src/variables.jl
+++ b/src/variables.jl
@@ -355,7 +355,13 @@ end
 
 Get a variable's name attribute.
 """
-name(v::VariableRef) = MOI.get(owner_model(v), MOI.VariableName(), v)::String
+function name(v::VariableRef)
+    model = owner_model(v)
+    if !MOI.supports(backend(model), MOI.VariableName(), MOI.VariableIndex)
+        return ""
+    end
+    return MOI.get(model, MOI.VariableName(), v)::String
+end
 
 """
     set_name(v::VariableRef, s::AbstractString)
@@ -1021,7 +1027,8 @@ function _moi_add_variable(moi_backend, model, v::ScalarVariable, name::String)
     index = MOI.add_variable(moi_backend)
     var_ref = VariableRef(model, index)
     _moi_constrain_variable(moi_backend, index, v.info)
-    if !isempty(name)
+    if !isempty(name) &&
+       MOI.supports(moi_backend, MOI.VariableName(), MOI.VariableIndex)
         set_name(var_ref, name)
     end
     return var_ref
diff --git a/test/constraint.jl b/test/constraint.jl
index a638154ec1e..5b8c26d98c9 100644
--- a/test/constraint.jl
+++ b/test/constraint.jl
@@ -986,6 +986,33 @@ function test_constraint_inference(ModelType, ::Any)
     @test obj.set == MOI.LessThan(1.0)
 end
 
+struct _UnsupportedConstraintName <: MOI.AbstractOptimizer end
+MOI.add_variable(::_UnsupportedConstraintName) = MOI.VariableIndex(1)
+function MOI.supports_constraint(
+    ::_UnsupportedConstraintName,
+    ::Type{MOI.VectorOfVariables},
+    ::Type{MOI.SOS1{Float64}},
+)
+    return true
+end
+function MOI.add_constraint(
+    ::_UnsupportedConstraintName,
+    ::MOI.VectorOfVariables,
+    ::MOI.SOS1{Float64},
+)
+    return MOI.ConstraintIndex{MOI.VectorOfVariables,MOI.SOS1{Float64}}(1)
+end
+MOI.is_empty(::_UnsupportedConstraintName) = true
+
+function test_Model_unsupported_ConstraintName(::Any, ::Any)
+    model = direct_model(_UnsupportedConstraintName())
+    @variable(model, x)
+    @constraint(model, c, [x, x] in SOS1())
+    @test c isa ConstraintRef
+    @test name(c) == ""
+    return
+end
+
 function runtests()
     for name in names(@__MODULE__; all = true)
         if !startswith("$(name)", "test_")
diff --git a/test/variable.jl b/test/variable.jl
index ab1d3347363..83550a7bfa7 100644
--- a/test/variable.jl
+++ b/test/variable.jl
@@ -990,6 +990,18 @@ function test_Model_inf_fixed(::Any, ::Any)
     end
 end
 
+struct _UnsupportedVariableName <: MOI.AbstractOptimizer end
+MOI.add_variable(::_UnsupportedVariableName) = MOI.VariableIndex(1)
+MOI.is_empty(::_UnsupportedVariableName) = true
+
+function test_Model_unsupported_VariableName(::Any, ::Any)
+    model = direct_model(_UnsupportedVariableName())
+    @variable(model, x)
+    @test x isa VariableRef
+    @test name(x) == ""
+    return
+end
+
 function runtests()
     for name in names(@__MODULE__; all = true)
         if !startswith("$(name)", "test_")