I agree AND disagree, I guess.
For the most part I'd say stick to one skill check OF THE SAME TYPE. In a pen & paper game you wouldn't typically go "roll Persuasion AGAIN to really convince the guy". You rolled already, it worked (or didn't). Of course there are situations where multiple rolls of the same type could also make sense.
I feel like having multiple DIFFERENT types of skill check (Persuasion, Religion, Arcana, etc) in a conversation can definitely be great so long as the devs remember that a success should equal at least some payoff, even if you didn't succeed through the dialogue chain.