When is an event so old that the phrase "the other day" no longer applies?

The Free Dictionary says "at a certain time past, not distant, but indefinite; not long ago; recently; rarely, the third day past." Collins simply says "a few days ago." So your girlfriend is closer to right. But to me, a limit of about a week, not a month, sounds right; otherwise, say "last week," "a week or two ago," etc.

Although most people agree that whatever happened last week, but not a year ago, can be said to have occurred "the other day", any attempt to specify time limits would be opinion-based. I might use the phrase for any event that occurred during the past four weeks, but some people may not agree.

This caught my eye, because my wife & I have very-very different perceptions of what "the other day" means. She grew up in SE Asia with a British school education, so she's fluent in English, so it's not a misunderstanding of the idea, I think it's a cultural perception thing.

When she says "the other day", it can mean any time in the past few years, whereas I, growing up in the US, think of it as no more than maybe a couple of weeks ago at the very most. This has lead to some interesting discussions until we realized the disparity in our views. I don't believe either view is wrong, just different.

This doesn't really answer the OP's question, it just sheds some light on how different people may perceive the passage of time differently and so the amount of time that has passed when the saying is used can have fluid boundaries.

Crystal wrote in 1966:

"Time relations in English are handled more by the careful use of adverbials ... than any other means."

To my ear, the word "day" anchors the past event to a point in time past that can be easily counted in days. Were you to say "the other week", I would consider the event to be easily countable in terms of weeks. Likewise for larger units of time: months, seasons, etc.

All we really know is that event happened in the definite past and, without other context, have to interpret based on individual associations with the phrase.

Citation

I would use "the other day" in place of a day that I could specify but the exact day was forgotten (or unimportant) - was it yesterday, or the day before, or last Tuesday? Once the event occurred at least a week ago and I'm scrambling to remember which week it was, not which day (I wouldn't usually specify "three Tuesday's ago"), I would use "the other week".

As noted by others, age may play a factor, too: at an older age, events that occurred within the past few months might seem as if they occurred within the week.