I’m posting this because I can’t find the issue documented anywhere, and I just wanted to let people with the same problem know they aren’t going crazy.
I don’t have a fix, but it happens to me often enough to be an issue.
[UPDATE] One solution (as posted by Greg in comments): find any disabled clips in your timeline. If two disabled clips are butted against one another, Premiere will not play (obviously a glitch in the program). Thanks, Greg!
I have had this problem when only one clips is disabled, but hopefully the above works for you!