Covems,
I have had the same question. Currently it is only possible to have 20 'open' messages in your inbox. Unfortunately the messages are not able to be copied+pasted. I personally missed out on a lot of messages because my inbox was full. I have saved messages as well, but decided at some point to delete them.
As far as I know, they have limited the amount of messages to 20, in order to keep server space to a minimum. As they are stored on their servers and not on your own computer. A solution would be if the sent messages were stored on your own hard drive. Only re-installing or deleting AChat would jeopardize the messages then.