Auto update and theme install not working

I can’t install new themes. What happens is that WP start unpacking, and then somewhere in the installation finds a bad excuse for quitting, with messages like
Can’t copy file /mydomain.com/public_html/wp-content/themes/”the theme I’m trying to install/images/bullets/diamond2-blue.gif

Sometimes it simply says
Couldn’t create “certain directory name”.

Not incidentally, the directory name in question tends to have the same name as my main directory on the server.

By the by, also just noticed that there seems to be some similar issue with the automated updating function.
If anyone here perhaps might share a few thoughts on where I should look for the problem, it’d be great!