Can, Could and Be able to

We use can to say that something is possible, or allowed, or that somebody has the ability to do something.The negative is can't or cannot.

• Maria can play the piano.
• Can you speak any foreign languages?   
• I'm afraid I cannot come to your party next week.
• I cannot climb a tree. 
• Can you open the door?
• Tom is only three but he can swim very well.

Be able to is a little more formal and less usual than can.

• We are able to see the lake from our hotel.
• The child is not yet able to write.
• I’m sorry that I wasn’t able to phone you yesterday.

But can has only two forms: can(present) and could(past).

• Abby can speak French, Spanish and English.
• My grandfather could speak five languages.
• He could run so fast in his childhood.
• I could swim when I was 4 years old.

But in some structures we always use be able to, not can.

• I haven’t been able to sleep recently.
• Jack might be able to come tomorrow.

For ability or opportunity in the past, we use could or was/were able to.

• My sister wasn't able to swim when he was 6.
• My sister couldn't swim when he was 6.
• Julia could / was able to  play the piano when she was five.

We use could especially with: see  hear   smell   taste   feel remember   understand

• I was sitting at the back of the theater and couldn't hear very well.
• We could see the village in the distance.
• I couldn’t understand what was happening