Okay, I figured it out.
I'm new to T-Mobile, so all I know about how to use it is what's on the phone. There is no 'Voice Mail' button, at least that I could find. There is a 'Visual Voice Mail' button. So that's what I used. And that's what led to the ridiculous situation where it was asking for a new PIN, but then told me that it wasn't the correct PIN. I would call this a bad user-interface software bug, and T-Mobile should fix it for future customers.
The solution is to set up 'regular' voice mail. That is found by pressing the '1' button. Maybe regular T-Mobile customers just know this, or maybe their phone has a 'regular voice mail' button, because otherwise how did other customers figure this out?
After doing this, and entering a PIN here, instead of under Visual Voice Mail, then Visual Voice mail worked!
(Thanks to T-Mobile support, who were very helpful despite the 30 minute wait. I hope the wait is because T-Mobile is getting lots of new customers, which increases the number of calls to their support staff).
By the way--I love T-Mobile--so much better than Sprint!!!