May, might, can, and could (possibility 1)

A. In affirmative sentences (that is, sentences that are not questions or negatives), we use MAY or MIGHT to say there is a possibility of something happening or being true:

This may/might be his last major speech before the election.

The news may/might come as a shock to many of the people present.

When Frank gets a job, I may/might get the money back that I lent him.

There is often little difference in meaning, but MIGHT can suggest that there is less possibility.

We can also use could, but not can, to express similar meanings. We prefer COULD to show that we are giving an opinion about which we are unsure:

'Why isn't Tim here yet?' 'It may/might/could be because his mother is ill again.'

There may/might/could be some cake left. I'll go and look.

We can use CAN in affirmative sentences when we talk about a more general possibility of

something happening rather than the possibility of something happening in a particular

situation:

The temperature can sometimes reach 35°C in July.

Mountain daisies can be yellow or red.

It may/might/could rain later, {not It can...)

We prefer MAY rather than can in more formal contexts:

Exceeding the stated dose may cause drowsiness. 

B.  We don't use MAY to ask questions about the possibility of something happening. Instead, we

use, for example, could(n't) or the phrase be likely:

Could it be that you don't want to leave?

Are you likely to be visiting Greece again this summer?

It is possible to use MIGHT in this type of question, but it is rather formal:

Might they be persuaded to change their minds?

C. In negative sentences, including sentences with words like ONLY, HARDLY, or NEVER, to say that something is not the case we can use CAN’T (or more formally cannot) or couldn't (or could not):

There can't/couldn't be any milk left - 1 would have seen it in the fridge.

There can/could hardly be any doubt that he was guilty.

Compare the use of MAY/MIGHT and CAN/COULD in negative sentences:

There are plans to rebuild the town center, but it may not / might not happen for another

ten years. (= It is possible that it won't happen for another ten years.)

There are plans to rebuild the town center, but it can't/couldn't happen for another ten

years. (= It is not possible that it will happen for another ten years.)

The difference is that we use MAY NOT or MIGHT NOT to say that it is possible that something is not true, and CAN’T or COULDN’T to say that it is not possible that something is true.

D. We use MAY WELL, MIGHT WELL, or COULD WELL to say it is likely that something will happen:

• The profits of the company may/might/could well reach $100 million this year.

We don't use can well in this way to talk about the future. However, can well is used to talk

about something we think or feel now:

• I can well recall how I felt when John told us he was moving to South Africa.

Other words commonly used after may, might, could, and can to say it is possible that something will happen are conceivably and possibly:

• The President may conceivably call an election in June. (= it is possible to believe it)

• The new parking restrictions could possibly lead to fewer cars in our cities.