"ORA-12520, TNS:listener could not find available handler for requested type of server" in JBoss EAP

Issue

The following error appears many times in the server.log and the database (with a 2000 connection limit) is running out of connections:

... IJ000604: Throwable while attempting to get a new connection: null: javax.resource.ResourceException: Could not create connection
...
Caused by: oracle.net.ns.NetException: Listener refused the connection with the following error:
ORA-12520, TNS:listener could not find available handler for requested type of server
        at oracle.net.ns.NSProtocol.connect(NSProtocol.java:395) [ojdbc6-11.2.0.3.jar:11.2.0.3.0]
        at oracle.jdbc.driver.T4CConnection.connect(T4CConnection.java:1102) [ojdbc6-11.2.0.3.jar:11.2.0.3.0]
        at oracle.jdbc.driver.T4CConnection.logon(T4CConnection.java:320) [ojdbc6-11.2.0.3.jar:11.2.0.3.0]
        ...