Great post Ant. I can relate with the 20 sax players and "Take The A Train" thing, only with bad guitarists and "Stormy Monday".
I went to blues jams because I knew I could play blues. If I went to jazz jams (and I should) I'd just be observing until I felt confident I could get through the material. So when you get there, just be a fly on the wall and evaluate to see if YOU like the atmosphere, players, music etc.
When you hear what's being played, you will either think to yourself....I can play this stuff, or I'm not too confident with this stuff. That's when you decide to stay there, or find another jam to evaluate. When you find a good one, be a regular, and play with whoever they stick you with graciously. (sometimes you get ahem, less than good players)
It takes some time to get entrenched in a good local jam but it's just the best investment of your time. Eventually you will get to be one of the A list players, and it all happens from there.
A few tips: People running the jam....usually a house band....it's considered courteous if you show up to support the house band. Go see their set. Watch them close, these are the guys you want on you side. Alot of the times they are the best players as well. I always thought it was rude to show up 3/4's of the way till the end of the jam, stay for a 1/2 hour, play, then leave. That's not a way to endear yourself to the players, socially speaking. I used to show up early, and stay till the end. Always. I would also watch every player closely, not just hang in the back listening and shooting pool. I was serious, I wanted to be an A player. When the house band asks you to sub, you're on your way.
I consider open mic blues and jazz jams to be a form of community college for musicians.