Yes, of course you can. With IMAP mail protocol both are always synced for the messages are stored on server and only copied locally; deleting a message on server deletes it on both clients next time they are connected.
Be careful with POP3 mail protocol, remember to select "Leave messages on server when read" when setting up the POP3 email on clients as most clients do delete the read messages on server by default. In your case this would mean that when you have read the messages with client 1 and later would like to read messages with client 2, they are no longer there.
I really do not see the point in using more than one client but it is possible.
Kari