I have done some more investigation and made some headway on this issue. I found errors in the system logs which correspond to the times when IWP fails. The message that appears is something along the lines of "DocWindow autoreleased with no pool in place - just leaking."
I realized in troubleshooting that I have 9.0v1 installed on the serving computer and that computer is running the Leopard OS. I think version 9.0v3 addresses certain IWP issues associated with Leopard, so I am installing the update right now. I am hoping this clears up this issue.
Definitely be sure you're running the latest updates. You should also really be serving off of FileMaker Server Advanced instead of FileMaker Pro, as it is *much* more robust. You should also consider upgrading to version 10, at least on the server end of things.
Note that there are a number of FileMaker hosting providers, such as ourselves:
which can provide you with the resources to host your databases off FileMaker Server Advanced without having to invest in the hardware, software, etc.