Hello TheShrevest and GeoFree,
Apologies to the both of you for the problems you're having with the WordPress upgrade. In this case it looks you're both having the same problem. It appears to be a temp file issue which is normally an indicator that some type of caching
is being done. If you are using caching, please make sure to de-activate the plugin first. Then run the upgrade.
If you continue to have the problem, checkout the following article: WordPress 3.6 upgrade bugs
. There are two links here that may give you an answer for the problem you're facing.
Apologies again for the problem and the delay in getting you an answer. Please let us know if you continue to require assistance for this issue. If you do need further help, help us by providing any further error messages, or additional steps in duplicating in the issue.