Hello Leosghost and incrediBill,
It's not possible to use another router. The one I've got is a combined modem/router Thomson TMG850, supplied by the ISP and is the only one that works with this cable connection. If I could, I've would have swapped it with the one from my neighbours. They're with the same ISP and have exactly the same modem/router.
For as far as I understand this matter, I too think the issue is caused by some setting on the server of the webhoster. The bad part is that this is my developserver, where I build websites for my clients.
I've rebooted everything twice, took out the powerplugs for more then an hour (also hoping this would get me an new IP address, as was suggested by the helpdesk of the ISP).
I'm almost certain it's not a problem from the computer itself. In this house are three computers, one iMac, a Macbook and a Windows laptop. They all have the same problem. When I took the Macbook to the neighbours and used their internet connection, everything worked fine again.
At the time the problem occured, I was working on a Joomla website, editing the CSS files. On top of that, I don't have admin-access to the server. It's a shared server, where I don't have any rights to edit anything releated to the webserver software. It's a 'resellers-package', where the only access I've got is with DirectAdmin.
Thank you both for your interest and the attempts to help me out!