Because of the restrictions on betting in America, US readers may not be aware of how it is over here.
Adverts for betting are prevalent, particularly in the advert breaks in the middle of a televised sporting event. The adverts will commonly offer promotional odds for a particular bet which they may expect to make a loss on in the expectation of future custom.
These bookmakers may decline or restrict service to particular individuals if
a) They believe them to only be taking advantage of promotions and not doing enough "regular" (i.e. profitable) betting.
or
b) They are winning "too much". This is more likely to happen where their wins cause the bookmaker to make an overall loss on particular events (i.e large long odds bets on unpopular events).
Once a restriction is in place the bookmaker will attempt to identify the individual through a number of methods including tracking cookies on their PC (Google for "iesnare"). UK legislation to combat both money laundering and problem gambling has made this easier for bookmakers by allowing them to require passports/proof of address etc. before processing withdrawals.
Part of the BBC Documentary "Britain at the Bookies" was about one such individual who had had their accounts restricted for being "too good" at identifying value in the advertised odds.
Personally I think there should be similar legislation to that around loans/credit where if a particular rate of interest is advertised it must be offered to the majority of applicants, with the same applying to odds in advertisements.