Auto updates failing

I can’t update 3.6.1 to 3.7.1, nor are themes updating at same site. I get this for WP update:

Unpacking the update…
Could not copy files.
Installation Failed

When I try to update themes, I get this:

An error occurred while updating Twenty Thirteen: Could not remove the old theme.

What’s up with this?

Domain is https://www.slscc.org; WP in subdirectory https://www.slscc.org/chaicoughski/ .

Thanks for any help you can provide!