No.  According to the C standard, identifiers in an enumerator list must be distinct from all other identifiers declared in the same scope.

EDIT:  After reading the C++ standard, it should seem that I'm wrong Tongue Out  If you declare your enum as enum class, you should be able to reuse names.  So it would look like this:

enum class A

enum class B

//... where you're using these enums

B someVariable = B::None;
B someOtherVariable = B::Value1;
A yetAnotherVariable = A::None;

Downsides:  you have to use A:: or B:: to access the enum's values and you can't treat your enumeration's values as an int or bool anymore (so int number = A::None and A enumValue = 1 will not work).

EDIT2:  enum class doesn't seem to be in the C++ 1998 standard...  your mileage may vary (it may or may not be implemented in Visual Studio; can't try right now).  Guess I should be more careful which document I pull up (I grabbed the Oct. 2008 draft of the C++ standard).  If it's not in VS, your best bet is to prefix your constants with some distinct prefix (like ANone and BNone).