did u use the changed files only zip? if so..exactly which files did u upload...if u dont mind.
I have root access to the server I manage. So I used the patch file as I find it easier to upgrade using patch. I am very used to patching code - why oh why aren't mods also available as patch files? I digress...
How I upgrade... I take a cp -rp of the production Forum code directory & call it phpBB2-upgrade, patch it using the command provided in the docs. Resolve any .rej files anomalies & make sure all template subdirectories have any subSilver patches applied.
Make sure the install & contrib. sub directories are present in the phpBB2-upgrade directory. Then:
mv phpBB2 /non-web-accessible-filesys/phpBB2-2.0.13-as-installed
mv phpBB2-upgrade phpBB2
then run the upgrade script from a web browser, rm the contrib & install directories & check is all ok in the admin panel & that users are happy.
Downtime for users was about 5 minutes, which given this is a 100% volunteer driven site (including me), seems OK.
Don't ask me how you do any of this without access to a shell, I wouldn't know where to begin. I live & breathe via SSH.
I hope this helps some,