@ onthepie: thank you. I didn't think it would be an issue for a test setup, but I guess someone could change the settings in such a way to allow for some really bad uploads.
@ joachim: I thought I head read through all the docs, but there are soooo many. Anyway, this is the message, with different values for 2592, that keeps on coming up:
'Fatal error: Allowed memory size of 33554432 bytes exhausted (tried to allocate 2592 bytes) in /home/httpd/vhosts/skygliders.nl/httpdocs/album/include/picmgmt.inc.php on line 439'
After trying many different uploads, it turns out that small pictures (with the largest size somewhere under 2400 pixels) are doing OK, bigger ones don't. The filesize seems to not be a problem (as the docs state, pixels rule over file size), as a 400KB picture failed where a 600 KB pictures succeeded.
Since I have full faith in Coppermine (and all of its loyal developers) and since I have uploaded 10's of thousands of pictures of all sizes in many different albums, I just tried and tried and stumbled onto one last thing: I put the setting of 'Automatically adjust pictures bigger than max width or height' to NO (this is translated by me to English - I don't know what it says in your settings, because I can't get my version to go back to English...).
And now, uploading seems to work as expected! Maybe it is just my server or my setup, but I can live without this one particular option.
Thank you both for your input and help. The debug mode got me to look in the right direction.