Cause
The cause of the "Additional instances cannot be launched" message, as well as the related error message "Transfer aborted: instance not found," is too heavy a load on the instance servers. When too many players are running instances at the same time, the instance servers can become unstable and prone to crashing. The rationale behind the messages is that players would rather wait to enter an instance and have it work perfectly, rather than enter an instance immediately and then have the server lag or crash.
Checking the Forums
When you encounter this error message, check the official Blizzard forums for your region. Another player may already have created a thread about the issue; if not, create a thread yourself to inform Blizzard staff of the issue and then wait for a response. Blizzard staff may post updates on the situation in threads on the official forums, as well as in messages on the left side of the game's login screen.
Finding Other Things to Do
"World of Warcraft" has a large amount of non-instanced content that remains available even when instance servers are not functioning correctly. Consider doing some quests, working on a reputation faction or even creating a different character and leveling it from the start.
Warning
Even if the issue persists for a while and you find yourself frustrated at your inability to enter instances, avoid creating hostile or insulting threads on the official Blizzard forums. These actions are frowned upon by Blizzard and may cause you to lose forum access or even have your account entirely suspended for a few days.