Service stops responding (Ubuntu 14.04)

I am running the .deb version of Subsonic 5.2.1. I have a strange issue where I lose the ability to create connections over new sessions, though previous sessions may keep working just fine. I have a PC running Chrome that can access the server; reloading works fine and all is good.
Meanwhile, on other machines, the service is inaccessible. I am running both HTTP and HTTPS. If I look at Wireshark on the Linux machine running the service, it looks like the connection gets to the point of making an HTTP request, then the server hangs. When I make an HTTPS request, the server hangs before completing the SSL handshake. In HTTP, the following packets are seen:
In HTTPS, the following is seen:
Is it possible that I'm seeing this kind of problem, related to some kind of hangup when Jetty tries to handle a new session?
Meanwhile, on other machines, the service is inaccessible. I am running both HTTP and HTTPS. If I look at Wireshark on the Linux machine running the service, it looks like the connection gets to the point of making an HTTP request, then the server hangs. When I make an HTTPS request, the server hangs before completing the SSL handshake. In HTTP, the following packets are seen:
- client -> server, TCP SYN
- server -> client, TCP SYN, ACK
- client -> server, TCP ACK
- client -> server, "GET / HTTP/1.1" (plus all the headers)
- server -> client, TCP ACK
- ... nothing else ...
In HTTPS, the following is seen:
- client -> server, TCP SYN
- server -> client, TCP SYN, ACK
- client -> server, TCP ACK
- client -> server, SSL Client Hello
- server -> client, TCP ACK
- ... nothing else ...
Is it possible that I'm seeing this kind of problem, related to some kind of hangup when Jetty tries to handle a new session?