Internet only works when connected via VPN

I've been having problems getting on the internet with my iPhone 4 (with the latest iOS) for a while. The "symptoms" are as follows:

- I get the "3G" symbol that shows me that I'm supposed to be connected to the Internet

- I can check mail and send mail

- I seem to receive some push notifications (but I'm not sure whether all of the ones I should receive get through)

- I can tether using the personal hotspot function - and all services on the computer work (http and others)

- but I can't connect to most other services (Safari, Facebook, BBC News etc.) straight from my iPhone

Once I connect to a VPN server, this restriction is lifted and I have full access to the Internet.

I double-checked all settings (eg. under "cellular data network") and as far as I can tell, they are all as they should be.

I tried re-installing the iOS and restoring from back-up - no change.

I exchanged the SIM-card with one from another provider - same problem.

Any hints as to what I (or the phone) might be doing wrong?

Sebastian