Internet Problems

I hate these. We have 2 or 3 machines networked then out through a BT 205 box. Internet is off.
Well it isn't really as 2 emails have crept through. Web browsing will not work.

The BT box says we are synchronised.

The dreaded phonecall. The person of course is of foreign extraction and English is not her first language. So she has Lisa try various ping tests. Looks impressive but shows nothing is wrong.
This goes on for 30 mins. So she passes us onto a different department. The man on this end says it will cost £25 if they fix it. We decide to decline.

So still no internet

Before ringing BT I have restarted the computer, turned off BT box for a few minutes.
So I have the idea of turning network box off.

Put this back on after a minute, hurray we have internet.

How does a network box allow 2 emails through but no browsing?
We keep a spare BT box adsl modem after years of problems.