If you are talking to someone on a quiet roadside - you can hear each other as you talk perfectly normally... moderate signal, low noise (high signal to noise)
Then a truck comes past - if you continue talking at the same volume, the noise will tend to swamp it, but you can still hear it if you concentrate. Moderate signal, moderate noise.
Now a phalanx of motorcycles comes past - if you continue at the same volume there is now no chance of hearing even if you concentrate. Moderate signal, high noise. (high noise to signal)
Similar with your DSL broadband - the signal level is essentially constant, if the noise increases it has to "work harder" (do more error recovery/correction) and the protocols ramp up the amount of error recovery/correction information sent per databit (which decreases the data capacity of the line - and also increases the latency).
The available bandwidth on a DSL line is divided up into "bins" - basically a slice of the frequency range. Each bin carries some bits - noiseless bins carry more bits that noisy ones.
You're right that as bandwidth is increased then the upper limit of the frequency range is increased but thanks to the laws of physics the higher the frequency, the higher the attenuation and that means the received signal is lower (and the noise proportionately higher) the higher frequency you use - so the higher frequency bins carry less datasimilar applies to long lines (and you lose the higher frequencies faster than the lower ones as the line gets longer too)
You can see both of the above here -
https://www.thinkbroadband.com/broadband/hardware/re... in the "Connection Tweaking and DSL feedback" section.
Add on to this other sources of interference - like broadcast AM radio - if you live near the Radio 4 longwave transmitter, and look at the spectrum of your DSL line there's a very obvious gap around 200kHz - which is not unexpected because R4LW is on a198kHz carrier. That puts those bins out of use and decreases the information throughput.