On the off chance that someone like me finds this thread in the future with the same issue, here's what happed to us and how we fixed it.
We have a phpBB system that we use for playtesting our games. It is basically on autopilot; none of us administer it regularly. We noticed recently that the system told us we should upgrade to 3.3.1 from a previous version. We clicked the links in the admin pages and the install failed (we got the "timeout error" for those of you scoring at home).
Because the install failed after starting we had a board in a highly misconfigured state; we had the software apparently upgraded to the new version but the database was still using the old version.
Here's what I did to fix:
1: Downloaded the most current stable version, unpacked it into it's own directory
2: Deleted all the contents of the "cache" directory in the current board directory
3: Changed the PHP version for our install in our host's Control Panel to 7.x, while activating the "sqli" plugin for PHP 7.x
4: Edited "php.config" in the root of our current board directory to change the database from "sql" to "sqli" (hat tip to the above poster who made this suggestion!)
5: Copied the install directory from the fresh download to the current board directory
6: Visited the upgrade page for our current install
7: Selected "database only" and completed the upgrade
8: Deleted the install directory from the current board directory
9: Appropriate thoughts and prayers to higher powers
Logged in. System works, now in current version for both software and database.