A couple of times one of our Macs has inexplicably dropped the Airport wireless connection to our BT Broadband HomeHub, and refused to rejoin it (even though the other Mac remained online). The error message when attempting to reconnect to the network says something like:
There was an error joining the Airport network BTHomeHub-XXXX [Try again] [OK]
I have the simplest of fixes: put a $ in front of your password
I'm taking a moment to document this here as I couldn't find this info anywhere else online. It's not in BT's troubleshooting guides or FAQs, and I ended up phoning their helpline for advice.
If you found this page when searching for a fix, please let me know if it works for you!