Yes I've had a 870 jam several times while hunting. I have missed several deer due to that gun not properly functioning. It's why I do not at all like the 870 Super Magnum Expresses any more. Or even the Express models really. That gun just left a bad taste in my mouth for 870's.

Other than that I don't think I've ever had a jam while hunting if I have it wasn't one that mattered much as I don't remember it. I did once have my 30-06 fail to go off but that was 100% my fault. It was a Browning BAR and I tried to slowly let the bolt go forward to not make much noise. When I pulled the trigger it just clicked. Turned out that while the bolt part was closed the part that slides forward from the bolt wasn't as I hadn't let it go forward hard enough like you need to and I hadn't manually pushed on the little hand lever thing on it to make sure it was all the way forward.