We have seen lots of strange behavior with 6.1.1. However 6.3 is a major revision, so it'd probably contain some new issues.
You might try upgrading to 6.1.4. We've been runnning it in several locations and it's been rock solid.
Also double-check that the IP address of the router isn't in any of your NAT tables...the Pix will proxy-arp for it's NATed addresses, which can cause interesting problems. We've seen Macintosh machines boot up with "Duplicate IP address" warnings because the pix was responding for their IP Address in a way they didn't like. (Never got around to troubleshooting it with packet capture, though).
One thing to try is "sysopt noproxyarp if_name". We've found this helps with strange ARP and/or IP address conflicts that we can't otherwise find a reason for.