Is Lt. Columbo psychic?

I like the first answer very much but I thought I'd provide my own take on the question with some of the cases in the show I could remember.

The best example off the top of my head is the episode Murder Under the Glass where the killer, who is also a renowned cook, poisons his victim when he dines with him. Columbo, after busting him at the end, tells to him that he figured out he is the killer as soon as he met him because he was surprised to see him arrive at the death scene immediately and not to see him go to the doctor after the person with whom he had dined died of food poisoning.

In another episode, Death Lends a Hand from Season 1, when Columbo investigates the body of the victim he finds the cut on her lip from a slap (and also the first person on the police team to notice that) and in the first scene where he is introduced to the killer by the victim's husband he says to him he can read palms and by that I think was able to make the deduction by looking at his big ring on his hand as soon as he meets him.

There were other instances like the episode Negative Reaction where the killer kills his wife and tries to frame another person by killing him in a ransom exchange which he schemed, but Columbo gets the facts from the scene of murder where he investigates a witness who claims to have heard the shots when he stated that there was a considerable time lapse between the shots and when he asks about it to the killer his justification pretty much isolates him as the prime suspect.

Also the case with The Art Collector where he was so focused on setting up his alibi at an art show after he kills his uncle by asking time from almost every key person in the art gallery which immediately puts Columbo right onto him.

If you look at all the Villains in Columbo they are mostly successful people in their profession, celebrities and most importantly incredibly intelligent that they cover up all the major loose ends and their alibis, but they get caught on a much simpler detail they might have overlooked which was exploited by Columbo and his dogged persistence.

The best quote I could think of coming from Columbo about himself its from Bye Bye Sky High IQ case where he says to the killer:

All my life I kept running into smart people. I don't just mean smart like you and the people in this house. You know what I mean... I could tell right away that it wasn't gonna be easy making detective as long as they were around by working harder than any of them, reading all of the required books and paying attention to every detail.

Columbo is by no means psychic. Did you watch the end of the episode? At the end of each episode Columbo always explains to the killer when he goes to arrest them what they did to tip him off, and why he kept pestering them. Something else you'll notice about all of the killers as you watch the episodes is that the killer always starts out trying to make excuses to get away from Columbo. You could probably say this is his first clue in all of the cases.

Another explanation might be that Columbo does in fact pester all of the other people involved, but the series just doesn't show him pestering those people too. I mean the episodes are already an hour long as it is. Can you imagine how long each episode would be if it showed Columbo pestering everyone?

In 'Columbo goes to the Guillotine' our eponymous hero specifically states that he's not psychic.

"I just wish I was psychic because this thing has still got me puzzled."

and

"If I knew what a suspect was thinking, I would make one terrific detective"

So there you go. Straight from the horse's mouth, as it were. Unless he's lying, of course...

Just to add to Dredd's answer, this quote from the original TV movie that started the series, "Prescription: Murder".

Lt. Columbo: Tell me, Doctor, how do you catch a man like that?

Dr. Ray Flemming: You don't.

Lt. Columbo: You're probably right. He sounds just too clever for us. What I mean is, you know, cops, we're not the brightest guys in the world. Of course, we got one thing going for us: we're professionals. I mean, you take our friend here, the murderer. He's very smart, but he's an amateur. I mean, he's got just one time to learn. Just one. And with us, well, with us, it's - it's a business. You see, we do this a hundred times a year. I'll tell ya, Doc. That's a lot of practice.

In all of the Columbo episodes, the murders are not acts of street crime like burglary or break-in, but carefully premeditated murders by individuals with very strong motives to kill. Columbo does not solve street crimes, like drive-by shootings. I think Columbo knows from experience that more murders take place by family, friends, business associates than by complete strangers, but are disguised as street crimes. As Cain murdered his younger brother, more murders are committed by close people -- like siblings, spouses and friends -- than by complete strangers.

Columbo is not psychic, but he has almost preternatural instincts. He hears or sees the very slightest inconsistency and gloms onto it immediately. That’s why he’s such a great detective. It’s attention to detail.

I think the writers plotted the episodes that way on purpose. The fun thing about watching Columbo is the way he and the killer interact. Why waste time getting to the best part of the show?