You can have 3, but hopefully you don't have too many other apps on your report. I applied for the US Air card last week, and they grilled me pretty good about why I had applied for so many cards within the past year. They're the only bank to have ever done that.
Then again, they only offered me a $7500 credit line, whereas all the other banks involved in my AOR pushed me over a total of 50k each (ok, one put me in the low 40s, but these were my first cards with them, so they gave me that amount as a new customer, on just two cards). In other words, Barclays is stingy, and a little more difficult. There's a reason the advice is generally to make them the first app in your AOR.