I have really tried to find a solution to my problem but can't.

When trying to run apt-get upgrade I get the following

root@anderslemke:/# apt-get upgrade
Reading package lists... Done
Building dependency tree... Done
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.
1 not fully installed or removed.
Need to get 0B of archives.
After unpacking 0B of additional disk space will be used.
Do you want to continue [Y/n]? y
Setting up at (3.1.10) ...
Starting deferred execution scheduler: atdinvoke-rc.d: initscript atd, action "start" failed.
dpkg: error processing at (--configure):
subprocess post-installation script returned error exit status 1
Errors were encountered while processing:
at
E: Sub-process /usr/bin/dpkg returned an error code (1)

Does anyone have a solution or explanation to why this is?

Sincerely
Anders

If you search a bit you'll see other people have encountered similar problems too (I've encountered one, but it was a bit different). Such errors are usually caused by a small bug or error in some of the scripts that should deal with installing or removing software, and sometimes you can even install some package which is impossible to remove. Apt refuses to do anything before it's previous job is done, so you'll have to deal with the problem before doing anything else.

You can try first to reinstall the package that causes the problem, after which you can try to re-remove it. Probably fails, but try. Then you should read apt's man page and pay attention to the "fix broken packages" option - try it out, if it works. Third option, which is often the only one that works, is to either fix the broken script/jamming part to make the process succeed or to do some manual work and remove the package manually (this doesn't mean that you simply remove the files, in addition to that you'll have to alter apt's files to remove the package's name from the installed list so that apt doesn't think it's installed anymore).

It's a difficult situation, I know, since the way of cleaning the mess differs depending on the package involved. Often the simple fixing methods won't work, you just have to fix it manually; some websites have information and examples on how certain broken packages can be removed (and/or installed and fixed), so I advice you to do a search on the web and read such pages through. First try the same package name you're having problems with, if somebody has already solved it, and if that does not help, try searching for more common terms to get your hands on the problem.

This sounds like quite a challenge for such a newbie like me. :-)

The biggest problem is, that I'm not quite sure which package is actually causing the problem.

The problem is mainly in the 'at' utility, as indicated by this line:
atdinvoke==atd invoke
initscript atd is self-explanatory

So, on reading the at man page, and a webpage (French) describing the same problem, it seems that at has a job in queue that it sees as incomplete.

You can list the queue with: atq -q a, or atq -q b. The a queue is the at queue, and the b queue is the batch queue. Since I don't know which queue is involved when running apt-get, I'd say list both to see if any jobs are pending. Jobs in the queue are listed with job number first.

Then, you can clear the queue with: atm <jobnumber>

Once the queue is cleared out, the problem should go away.

Well, I removed at (apt-get remove at) and that seemed to solve the problem... :-)

While running at, there were no jobs in either queue a nor b.