You don't have to carry your passport on you. A) They only stop people who 'look suspicious' (racial profiling, anyone?); B) If you sound confident about your status and your facts, they'll likely not suspect you; C) All it takes is a phone call to confirm your status. Carrying your passport with you is a bit silly, as the risks are far too great.
To me, watching the show just makes me laugh...it's not scary at all if you're a legitimate entrant with legitimate reasons. I've not seen one episode where they've turned someone away that actually sounds 100% legitimate. Everyone I've seen rejected has been quite rightly so...and sometimes they're quite generous with people they let in - some have very tenuous circumstances and still get stamped through - quite luckily.