I think many people use breakdown to mean their car has broken down.
Usually the context makes it clear, however, sometimes it can be ambiguous such as:
"I had a breakdown yesterday". This could mean your car broke down, but it could be that you were very upset and went into "meltdown".
It always makes me chuckle when I see something saying "Breakdown assistance".
Personally, I prefer to distinguish between the two:
For cars, or general machinery, "My car broke down yesterday" or "My car has broken down".
As opposed to:
"I had a [nervous] breakdown yesterday" or "I'm having a [nervous] breakdown".
Sometimes people will make it absolutely clear what they are talking about by using the term "nervous breakdown", then there is absolutely no confusion.
But as I said, usually the context makes it pretty clear.
However, breakdown in the sense of "They broke down and cried", to me, just implies that they were very upset for a period of time, because they'd lost their job, or a family member, or received other bad news. (i.e. they were not having a full nervous breakdown, from which they may not fully recover).
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