This could be a number of things. I used to do Tier2 support for DSL though and found the most likely cause for this is loss of sync.
Unfortunately you will often get techs on the line that don't want to troubleshoot or don't really know how to. If the only resolution you got was that your line is ok, they didn't do much.
(Also, if they did not have you unplug everything from your phone lines except the DSL modem, they did not check your line completely.)
You should have a sync light of some kind on your modem. It could say sync or internet or any number of things. Check and see if it is blinking when you cannot get online. If so; something is causing interferance with your DSL. There is a long list of what this could be.
Most common causes:
1. A phone, answering machine, fax, etc is plugged into a phone line but does not have a DSL filter on it.
2. Your DSL modem is plugged into the supplied DSL filter but does not share the filter with another phone device. (The DSL only has to be filtered if anothe phone device is sharing the same jack; otherwise plug it in directly.)
3. Something electrical (or wireless) near the modem is causing interferance. Move anything that runs electricity as far away as possible. Move Cordless phone bases a minimum of 15 feet away.
There are more things to check for sync; and other things that could keep you from connecting. If this doesn't work; comment with your DSL modem, DSL provider, and ip address and we'll go from there.