Basically means that your DSL modem is not seeing anything at the other end of the 'phone line.
More than likely a 'physical' problem - I.e. faulty cabling, line fault. microfilter problem or even being disconnected at the exchange.
If the AOL diagnostics tell you it is 'training' it is basically hunting for a signal but can't find one. That suggests that the modem itself is OK - the symptoms you describe are very unlikely to be caused by a modem fault.
Need to report it to AOL even if the line is provided by BT.