The only way they could do anything would be a lay a new telephone cable from your exchange to your home, however this is a major and very expensive undertaking and so is not really an option.
Indeed that would never happen lol. The most that could ever be done would be to try and trace out some spare pairs on existing cables and that only applies to lines which are struggling to get ADSL at all.
With a 5dB noise margin it would be expected that the exhcange monitoring equipment might aim for a higher target SNR margin. This means a slight drop in your connection speed in order to achieve a more stable connection. As the folks on here will say, the target SNRM is normally around 7dB.
With regards to a BT "health check", this normally involves a quick process using software whereby the exchange reports back a few simple stats on line resistance etc to check that all is OK with the line.
Indeed somehow having your maxDSL profile lowered would likely bring you a higher SNR M and some stability.