We flew out of Barcelona after a
Disney cruise two weeks ago. My mom was with us on the cruise, her flight left at 12:30, ours left at 2:00pm, we weren't in any particular hurry to get off the ship, took a Disney transfer from the terminal to the airport, got to the airport and found the right counter, waited to check bags and stuff, then went through security, grabbed a snack at McDonalds and walked out to the gate. We were all at the same gate by 11:00.
If we actually had to be there by 10:50 I think it would be doable. Disney makes provision for people needing to walk themselves off for early flights (might need to carry your own luggage instead of letting the hosts take it the night before). If you do that I think you can be off by 7:30am. If you had a prearranged private transfer or waiting taxi you could probably be at the airport by 8:00am. Then you need to take your chances with the check in and security lines.
I believe it may be technically possible but I wouldn't recommend it and believe it would be really stressful and harried.