At the time ?

I first met him in 2005. At that time he was a student at Harvard.

I started to quickly search through his desk. At that moment he entered the room.

Thank you for your inquiry. At the times mentioned in your letter, our records show that Mr. Smith was not at work.

What I'm confused about the word 'at the time' is its meaning in these contexts.

There was a parade going by at the time.
I was in fact thirty-one at the time.
Mr Brown was sick at the time.

I think I still don't get it precisely.

Thank you.

In this type of sentence, with the phrase "at the time," one could look at it as an example of ellipsis, which is very common in English (if something just doesn't seem to make any sense at all in English, it is often because words have been left out and are to be understood from the context or from the tradition of usage). For example:

He tried to cross the street but could not, becaus

To go back to your first post, Norpriest:

'At the times' is not a collocated phrase, though the words can appear in such limited situations as Anon as illustrated. It does not mean 'at the time' or 'at the moment'.

At the time = at the same time
At the moment = right now; currently and temporarily

Thank you.

Now I get it !