How come Col./Dr.Bellows always drove up to Capt./Maj.Nelson's house,and went in to the house uninvited?Was it to check up on Capt./Maj.Nelson and Capt/Maj.Healy?

He does that throughout all of the seasons, other than a handful of times when he rings the doorbell. As for why he does it, I have no idea, just some guesses.

Perhaps because he's Tony's superior officer, he feels entitled to be able to just walk in whenever he wants, as if Tony's house is no more than military barracks.

Or, it's as you said, and he thinks he has to check up on Tony ever since his admission in the pilot of finding a djinni in that bottle. Now Tony is forever suspect of mental problems by the good doctor, and thus his justification for prying into Tony's private life however he can.

One thing to notice in sitcoms all across the decades is how it's a rare occasion when they lock their doors at all, such that anyone can just walk in. I know people in other countries find the locking of our doors to be an alien concept, and further cementing in their minds that Americans are all paranoid. But, one time I pointed out to a native of Australia that even though neighbors are welcome to come in when invited, when we want privacy, you can bet the doors will be locked. They finally understood the privacy angle ;)

Which is why it's so perplexing that Tony doesn't lock his doors, considering what he has to keep secret! :)

How come Col./Dr.Bellows always drove up to Capt./Maj.Nelson's house,and went in to the house uninvited?Was it to check up on Capt./Maj.Nelson and Capt/Maj.Healy?

Yeah, I find it an invasion of privacy. At least in Bewitched, Larry Tate never did such a thing. Dr. Bellows needs to learn how to respect one's private life and knock. If someone did that to me, I'd get pretty annoyed.

...Dr. Bellows entering Tony's house "uninvited" was started when Sidney Sheldon began writing the majority of the scripts in early 1966. Apparently, he thought it would be funny if Jeannie put "Master" in an awkward position- in his own living room- just before the good doctor walked in, unannounced.

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