You can have at least 4 total Barclays cards, if you don't have a particularly high credit limit on any of them. I've had the Best Western and Choice cards for years, got the US card a year ago (and still have it though I'm about to cancel it), and was approved for the Wyndham Rewards card in December. I think I have about $5k on each one. (I have a very high credit score, though what it is exactly varies quite a bit from bureau to bureau, and to a smaller bit from month to month.)
I was turned down around November for a second US card, but the reason given was simply that I already have one US card and they won't issue me a second one. (Ie, it had nothing to do with how many total Barclays cards I had.)