I doubt they say they can improve your speed. They will attempt to fix your line if it's broken, in some sense that presumably you would have to come to agree upon, and give you your money back if they can't get it sorted. So AA are brave enough to actually want to take on customers with problems, as they really do try to sort BT etc out.
They can only do so much though, because if they can't persuade eg BTOR to do anything (by leaning on them) then you are out of luck. This could include situations where they know in advance that BTOR won't be willing to take action. Just so you know that their god-like powers are limited, of necessity, until such time as such small ISPs can gain more control over BT (in an ideal world).
There are also a number of things which they (and we) can try that might well improve speed and reliability, involving possible changes to wiring, change of modems and filters. You may very well have done a lot yourself, but otherwise putting in some time and money can sometimes give substantial improvements, as in my case.
"Bonding" : With AA you can also double, triple, or more, your internet connection speed if you can order more copper lines, because AA can combine them into a single n-times speed pipe. This is what I have done. It requires (i) cash, and (ii) being lucky enough that AA can order extra physical lines (from BTOR - or is it BTW or Talktalk Wholesale) without being landed with a horrendous connection charge, and (iii) the right kit at your end. This isn't usually a problem for one additional line afaik. I have three lines combined, and it really does give triple speed individual downloads, and also uploads with the right kit. I think there are other ISPs who can do this too. AA can even do this for you using lines that you bought from someone else, that is, not from them.[Moderator edited to merge three consecutive postings into one.]