Depends what product you're on.
If you're on a BT Wholesale product (either ADSL or FTTC) eventually BT's DLM will notice the instability and will attempt to correct it using a number of methods.
If you're on ADSL (or ADSL2+) this correction will most likely be in the form of an increased target margin, which will force your router to connect at a lower speed.
It's clear, from what you say, the default target margin (of 6dB) isn't giving enough capacity to cope with high noise periods (hence your margin is dropping to zero and triggering a resync).
If you're not on a BTw product (e.g. you have an unbundled product such as Sky, BE, O2, Talk-Talk, etc.) they don't all have something similar to BT's DLM (which most people are glad of, because BT's DLM sucks).
e.g. If you're on BE or O2, you'll most likely have to call support and ask them to increase your target margin/SNR to 9dB. Certain BE packages allow users to select this via a web-page.
vDSL2 FTTC Infinity with BT
DL Sync 80Mbps
UL Sync 20Mbps