screen - No more PTYs

I have a recurring issue on two SLES 11 servers where users cannot create screen sessions. Screen returns: "no more PTYs"

I say recurring because remounting /dev/pts fixes the issue--for a week or two.

I've found people on various forums reporting the same issue but they all had incorrect permissions for /dev/ptmx or were mounting /dev/pts with mode=0600. Neither of those seems to be the cause of my problem:

ls -la /dev/ptmx
crw-rw-rw- 1 root tty 5, 2 Jan 21 12:48 /dev/ptmx

mount | grep pts
devpts on /dev/pts type devpts (rw,mode=0620,gid=5)

I've checked the permissions on ptmx and the mount options when the problem is occurring and they're exactly the same.

I'm hoping someone here will have suggestions for a persistent fix.

EDIT

The problem has recurred this morning on one of 2 machines.

Max ptys is set to 4096. The really weird thing is only 23 people have logged into the machine between the last time this occurred (yesterday) and now, and there are no running screen sessions.

Strace didn't reveal anything terribly useful--at least not to my eyes: http://hastebin.com/enifumivow.vhdl