John, this sounds like a tricky issue.It's difficult to say without actually seeing your router configurations. However, I'm curious to exactly why you would like to have both routers active at the same time? If nothing else, having both routers active will certainly cause at least minor performance issues from the processing delay. You would be able to have multiple devices connect to a single router.
As mentioned, the issue could be related to many things, not limited to ip addressing issues, Network Address Translation, connection configurations, etc. However, we do know that your Internet is good, like you mentioned.
If you are attempting to use the Linksys router for its features or something similar, it's best to put the 2wire router into bridged mode. In short, this will pretty much make your 2wire router invisible to the linksys router and it will just work as a modem.
Here is a link I found to instructions on how to change your 2wire router into a bridge, although I'm not sure if this will serve your model:
http://www.dslreports.com/faq/15835
You could do search online with your router model and the keyword bridge.
Hope that helps, good luck!