Hi vfaulkner,
You can recover files from backup by opening a new .rp file and navigating to “File > Recover File from Backup” from the main menu of Axure RP. Axure RP automatically saves a copy of your working file every 15 minutes, dating back to 30 days, so if you’re unable to find any backups, your backup settings may have been turned off. If so, you can enable them again by navigating to “File > Backup Settings” and selecting the “Enable Backups” checkbox.
Regardless, if you’re unable to open .rpprj files, backups may not be of use. From the error message in your original post, the most likely issue is that Axure RP is unable to connect to the SVN team repository. If your other team members are able to get or check in/out of the team repository, please have them send you the SVN team repository address. They can grab it by opening up their own local .rpprj file and navigating to “Team > Browse Team Project History”. Please have them copy & paste this address over to you, and then try grabbing a new, local copy of the TP by going to “Team > Get and Open Team Project” and pasting the team repository address in the Team Directory field under the SVN tab.
If you’re still having trouble after the above, please write into support@axure.com, and wen can continue troubleshooting through there. Thanks!