If you navigate to the well-known developer Q&A site, Stack Overflow (www.stackoverflow.com), and run a search query for connect on the cqlsh tag, you'll notice something astounding. About 25% of all cqlsh questions are about people having problems using it to connect to their cluster. Given this prevailing issue, a discussion on connecting to cqlsh is warranted.
There are three common scenarios in which you might be trying to connect to your cluster, and how cqlsh will show you a failure.