The Doctor's success seems to be a combination of several factors. Intelligence may be one of them. I didn't find it yet, but I remember a conversation he had with the Master about their school days; I believe they were both at the top of their class.
In any case, there is some evidence for him being of high intelligence, but he is not the only one.
In "The Deadly Assassin":
ENGIN: It would require a mathematical genius with a phenomenal grasp
of applied excitonics.
SPANDRELL: Really? There can't be many of those on the High Council.
SPANDRELL: What is the Master like on mathematics?
DOCTOR: He's brilliant, absolutely brilliant. He's almost up to my
standard. What's that?
In "The Invasion of Time", Borusa (formerly his teacher), seems to be able to keep up with him, and when let in on the 'secret' inside the lead-shielded room, very quickly comes to understand what the Doctor is about.
Some of the Doctor's success is due to being unpredictable and creative. Even the Master says (in "The Deadly Assassin"),
He remains as ingenious as ever.
In "The Invasion of Time", Borusa and the Doctor discuss some differences in how they think, which is also enlightening:
BORUSA: And you did not think that I could shield myself?
DOCTOR: Well, Lord Borusa, you and the other Time Lords are singularly
logical, hmm? You're also short on humour and imagination. You can't
offer distractions as I can. What's for tea?
DOCTOR: See, you're too single-minded. You're as transparent as good,
old fashioned glass.
BORUSA: You're right. I wouldn't last a moment. My mind is too easy to
read. The master learns from the student, eh, Doctor?
And some is experience. Most Time Lords are not working in the field. Romana traveled with the Doctor, in part, to learn/benefit from his experience.
Many of the Time Lords, though, were simply completely sheltered, and thus unable to deal with many and varied challenges. This is very clear in the episode, "The Invasion of Time".
(Leela and Rodan make their way across sand dunes under an orange
RODAN: Oh, I'm so tired. I must rest.
LEELA: We haven't come far enough.
RODAN: I never thought it'd be like this.
LEELA: What, you've never been outside the Citadel before?
RODAN: No, why should I have? There's everything we need inside.
LEELA: This is much better than inside.
RODAN: It's frightening.
LEELA: Why are you scared?
RODAN: It's all so (pause) natural.
LEELA: We must go on. They can still see us here from the Citadel.
later, by outsider's camp
NESBIN: Protection. Help. You can't survive out here without.
LEELA: I can survive anywhere.
NESBIN: I believe that. What are you?
LEELA: I am a warrior of the Sevateem.
PRESTA: An alien! That's dangerous. Surely they'll come hunting for
NESBIN: We'll consider that in a moment. Well, warrior, I don't doubt
that you can survive, but what about your friend here?
RODAN: What about me?
NESBIN: Well, you're no alien. I doubt if you've ever set foot outside
the Citadel in your life before, have you?
NESBIN: Well, out here it's different. You have to fend for yourself.
NESBIN: That's right. What are you going to eat, for instance?
RODAN: I have some supplies. Look. (Some tablets in a box in her belt
NESBIN: They won't last long. When they've gone, what will you do?
RODAN: I, I don't know.
NESBIN: I thought not. You wouldn't even know what you could eat,
would you? Have you ever eaten flesh or fruit?
NESBIN: And shelter. Do you understand the need for shelter? No, of
course you don't. You wouldn't last three days out here.
RODAN: I didn't realise. I'm so tired and cold.
NESBIN: Better get her inside.
NESBIN: Oh, but we are, or rather, we were until we decided to drop
LEELA: What is drop out? You fell?
NESBIN: All that peace and eternal tranquillity. We decided to get
back to nature out here.
LEELA: Is this true?
RODAN: Well, I've heard it rumoured, but it's a subject that's never
NESBIN: Well no, it wouldn't be. Might upset their cosy little world.