at the time/ at a time


The phrases have different meanings and cannot be substituted for one another.

I didn’t think anything about the event at ___ time.
Here only “the” can go into the blank, the sentence is refering to a specific time (the time of the event).

Once upon a time,…
Here the time specified is unspecific and is actually likely to take place in an unreal world because this is a stock phrase used to start a fairytale or child’s bedtime story.

The phrase “one at a time” means not more than one at once. A cashier at a grocery store is likely to serve customers one at a time.