"org.apache.commons.dbcp.SQLNestedException: Cannot get a connection, pool exhausted" in JBoss EAP

Issue

Encountering DBCP exceptions such as the one below

... org.apache.commons.dbcp.SQLNestedException: Cannot get a connection, pool exhausted
 at org.apache.commons.dbcp.PoolingDataSource.getConnection(PoolingDataSource.java:103)
 at org.apache.commons.dbcp.BasicDataSource.getConnection(BasicDataSource.java:544)
 ...
Caused by: java.util.NoSuchElementException: Timeout waiting for idle object
 at org.apache.commons.pool.impl.GenericObjectPool.borrowObject(GenericObjectPool.java:756)
 at org.apache.commons.dbcp.PoolingDataSource.getConnection(PoolingDataSource.java:95)
 ...