When they say specific "time", they mean "occasion", "instance". So perhaps your car broke down on Monday, Wednesday and Friday of this week and you are talking on Saturday.

"This week" is an expression used for the current week, and so in this sentence, the time period is from last Sunday or Monday until now.  It's possible that the car will break down again this week. So the present perfect is a the most correct option here. 

From what I understand, in everyday conversation in many parts of the USA, the simple past would be used in this situation more often than the present perfect, but the present perfect would still be correct.  In the rest of the world, only the present perfect would be correct in this situation.

In response to Stephen Larrabee (thumbs up even if only to cancel out some unjustified thumbs downs), Dan from Massachusetts has debated this issue before with me and other US teachers, and he takes the British approach to this particular usage of the present perfect (repeated times).  So based on what he says, there are some regional variations in the US. However, I report his only as hearsay.

If I talked about "this week" on Saturday to mean the finished working week up to Friday, I can imagine using the past simple naturally too.

"We can't use Present Perfect with a specific time, right?"

Not exactly. We can't use Present Perfect with a specific time in the past.  Present Perfect already has a specific time: now.

When we use "this week", "this morning", "today" and so on with Present Perfect, we are still in that timeframe.  My car has broken down three times this week means in the timeframe starting at the beginning of this week, up to now (and still in this week).