"She's had such a hard time" means that life was difficult for her for a long time. Maybe she was born with poor health, maybe she had a bad accident and it took a long time (months or years) to recover.  A hard time is long and difficult with a lot of struggles before life is better.  Charles Dickens wrote a book about poor families in England titled "Hard Times". The economic depression in the 1930's caused misery and hard times for many people in many countries. 

"A rotten time" is something very different.

It can mean that you went to a party and it was not fun. Maybe people were teasing you or making fun of you. Maybe you saw your girl (boy) friend with somebody else. Maybe nobody would dance with you.

It can also mean that you went on a holiday and the weather was terrible, and you caught a cold and had to spend the holiday in bed.

I use tough time when a person has something to do and has encountered or will encounter problems. 

I suggest that when you would like to see examples on how phrases like "hard time" "tough time", are used, that you look at the examples in this database:  http://www.americancorpus.org
 

Studying quotes gives a good feel on how phrases are used, after you look at what the dictionary has to say.

Here are 3 examples out of over 800 hits for "tough time"

folks that are out of jobs, having a very tough time of it and unemployment at a high
Developers will have a tough time getting pending projects built, said Bill McCubbin
Clinton also had an exceptionally tough time during the campaign explaining the evolution of his belief system.

You can also see patterns, a large number of hits in a context, such as the expression "give someone a hard time" - meaning to challenge them on their behaviour or opinions.
As soon as I saw my young aunt, I knew she was a spirited woman who would give my uncle a hard time.