Can (отрицательная форма cannot или can't) употребляется в следующих случаях:

а) для того, чтобы сказать, что нечто является возможным:

• I can't spell very well.
• Swimming after eating can be dangerous.
• Making mistakes can be a good way of learning.
• Smoking cannot be good for you!

b) для того, чтобы сказать, что кто-то знает, как сделать что-либо:

• My brother can drive.
• Can you speak French?

с) для того, чтобы сказать, что кто-то способен сделать что-то:

• She's a great driver: she can drive almost any car.
• I can't eat fish.
• Anyone can become a qualified teacher.

d) с такими глаголами, как see, hear, feel, smell, remember, recognize, imagine:

• She can't remember the name of the book.
• I can't see the reason for doing that.
• Can you imagine living in a palace?

Вместо can может употребляться форма be able to, кроме случаев, обозначенных в пункте 'a', но модальный глагол сап является более употребительным и менее формальным:

• Don't shout! We can all hear you.
• My brother is able to drive. I am not able to eat fish.
• She isn't able to remember the name of the book.

Could (отрицательная форма could not или couldn't) употребляется в следующих случаях:

а) когда мы говорим о чьей-либо способности в прошедшем времени:

• Не could run faster than any of us.
• She could tell the most i ncredible jokes.
• A lot of them couldn't read or write.

b) с глаголами see, hear, feel, smell, remember, recognize, imagine в прошедшем времени:

• You could see they weren't happy.
• The policeman could smell gas.
• He couldn't see them, but he could hear them in the dark.

с) когда мы не совсем уверены в чем-либо:

• There's a lot of traffic. That could explain why he's late.
• There could be a storm tonight: look at the clouds!

Be able to употребляется вместо could:

а) после другого модального глагола {will/must/might):

• I might be able to help you later on.
• You should be able to buy some cheese in that shop.

b) если мы хотим употребить форму с -ing или to infinitive:

• It's nice to be able to get some exercise.
• He complained about not being able to go to London.
• I enjoy being able to get up late at the weekend.

с) если мы говорим о ком-либо, кто справился с какой-либо ситуацией в прошедшем времени (отрицательная форма wasn't/weren't able to или couldn't)

• Were you able to buy everything on the list?
• They were able to save enough money to buy a car.
• I wasn't able to finish the meal. (=I couldn't finish it)

Collins Cobuild Grammar