Application Servers locking Oracle Database Client libraries causing PeopleSoft to lose its connecti

This is a really odd problem.  3 times in the last 8 months (so it's pretty rare) we've had both of our app servers (two separate servers) lock up with the PSAPPSRV processes not able to create a connection to the database.  All existing connections work fine, it's just the new ones that fail.  This means that eventually (as all of the app server processes recycle) we lose our connections to the database.  At this point, all other network connections work including tnsping.  However, sqlplus connections hang before they even prompt for the password.

So far our only solution is to reboot both servers and that has cleared up the issue.