The problem:
You login to Sharepoint (WSS or MOSS), then click the "Sign in as a different user" drop down. You attempt to login using the same account with the same privileges (obviously) and for your efforts, you get a 401 Unauthorized message. What?
The fix:
DON'T USE THE SAME ACCOUNT
Seems rather intuitive, although the error message is not. "Sign in as a different user" really does expect a DIFFERENT user. In testing mode, it's common for developers to test and retest using the same account. Why not?
Incidentally, the same is NOT true for a simple Sign Out. Assuming you are using FBA, you can log right back in using the same account.
Hopefully that saves some hair pulling.