- They had a problem on five different occasions.
Was it necessarily the same problem?
- They had problems on five different occasions.
Could they have had one problem on each occasion?
Did they necessarily have multiple problems on all of those occasions?
When speaking informally, amongst friends, we don't always speak precisely, we don't always fully qualify every detail of what we are saying. In attempting to be more precise we run the risk of being boring and pedantic.
If what we are saying has some level of ambiguity and the detail becomes pertinent later in the conversation then we may add clarification.
Your suggested phrases may have a natural meaning, but as your question implies, leave room for some different interpretations. If we are trying to give a summary or feeling of a situation we deliberately remove detail. If the detail matters then we can choose to be more precise.
The railways in the UK are very reliable.
Only this month I had a problem on five different occasions
In this context it probably doesn't matter whether it was the same problem or different problems. It would be possible to be more precise, but it's not necessary for the sense of the conversation.
The railways in the UK are very reliable.
Only this month on two occasions the train was cancelled, once it ran but with only half the capacity, once it failed to stop at my destination and once it was an hour late.
not ambiguous, but very boring! If it were the same problem on the five occasions I would probably say
I had the same problem on five different occasions
So I would guess that your original sentence does imply more than one type of problem, but if it actually mattered I would seek clarification.
A strict interpretation of this:
They had problems on five different occasions.
would imply that on each occasion there was more than one problem, however I would not feel it was misleading if the count had actually been 3, 2, 1, 2, 1. It might be more precise to say
We had at least one problem on five occasions and, in fact, multiple problems three times.
again, more precise, but probably needlessly so.
