Why does Time Machine say it's unable to back up because the disk is full -- even tho it's supposed to delete the oldest backups when that happens?

I'm running OS 10.6.8 on an imac from around 2011. I back up with Time Machine to an external hard drive that has about 500 GB. Until now, I've had no problemsm, or at least that's what I thought. But today I'm getting a message every hour saying "Time machine is unable to back up because the disk is full." It IS full, but Time Machine is supposed to delete the oldest backups when that happens, and keep deleting them in order.

Anyway, when I opened up my backup disk I discovered backups from February and most of March, nothing in April and nothing in May.

I'm a little freaked.

Does anybody know why that would happen and what to do about it?

Thanks in advance.