You can send out as many cultural exchange offers as you have spots open in your museum. That means you have 11 museum slots and can send out offers until all 11 slots are full.

You can accept an unlimited number of cultural exchange offers regardless of the condition of your museums. If you have 26 cultural exchanges going and only 1 level 1 museum then you have 25 extra, but still valid, cultural exchanges.

Basically people who ask for cultural treaties are wasting their breath because people with maxed out museums can't send out offers and can only accept them. Therefore unless they take the initiative and send out the offers themselves such requests are only valid between people looking for exchanges in the first place.

It is best to just send out the offers as you get them to random people in the alliance (or all of them for that matter) when you have to power to do so and stockpile a surplus of cultural goods. It harms no one and does everyone a world of good.

There is one little issue though. If you have exactly the amount of goods needed to fill your museums you will not be able to accept more exchanges through your diplomacy adviser. You have to go through the museum in your town and manually accept it with the drop down menu. If you have a surplus of goods you're once again allowed to accept all treaties normally.