Jenkins 1.570 (and many previous versions) stop responding to any HTTP requests after running for somewhere between 5 minutes and an hour. Increasing the heap size didn't seem to help, and jmap -heap seems to indicate that there is enough memory available.

jstack -F (jstack failed without -F) seems to indicate that all the threads are blocked, but I can't tell why Jenkins doesn't respond to HTTP requests, or even if there is anything I can do to fix it other than restarting the entire Java VM.