“I've been known to have 5 or 6 cups in a day.”

You told your friend that you were trying to cut down on caffeine, so you started talking about how much coffee you drink. Sometimes you drink 5-6 cups of coffee in one day. You say this to your friend.

I've been known to have 5 or 6 cups in a day.

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(something) has been known to (do something)

Use the phrase "___ has been known to ___" to describe something that sometimes happens or has happened a few times in the past:

They 've been known to rehearse 10 hours at a single stretch.

I'm not sure if I trust you. You have been known to make promises and not follow through on them, you know.

The phrase "__ has been know to ___" can either sound very careful, or sound funny and playful. So you could use it to carefully talk about mistakes your boss sometimes makes:

She's been known to miss important client meetings from time to time.

Or you could use it to joke about something embarrassing that you do a lot:

Yeah, I've been known to buy a new pair of shoes every now and then.

The joke here would be that you actually buy new shoes all the time, and it's funny that you're trying to sound like you don't do it a lot.

have (a drink)

English speakers often use the phrase "have ___" instead of "drink ___". For example:

Do you want to have a glass of wine with me?

You use "have a drink" when you're talking about the experience of drinking something, or you're talking about drinking in a social situation.

If you're describing what you drank in a scientific- or medical-sounding way, you would use the verb "drink" instead of the phrase "have a drink". For example, if you got sick while on vacation in a foreign country, you might describe what made you sick:

I drank some fruit juice with ice in it on the second day, and I think that the ice is what gave me the stomach virus.

cups (of coffee)

"Coffee" is a liquid, so you can't count how much of it you've had. The way that English speakers count coffee is to say how many cups they've had.

Hot tea is counted in the same way. Cold drinks are counted by how many "glasses" or "bottles" you've had. Beer is counted with "beers":

I'm totally not drunk! I only had, like, two beers!

(number) in a day

If the speaker had said "5 or 6 cups a day", it would have sounded like that was how much she drank every day. But since she said "5 or 6 cups in a day", the sentence means that there are some days when she drinks 5 or 6 cups. On most days she probably drinks less.