You can either get this intentionally or accidentally. When you select a city from the sailing screen by touching it, you will be presented with the option to enter or bombard and you will be shown the number of cannon protecting the city as well as how much gold is in the city. The number of cannon is very important.
Also, the cannon limit may or may not relate to your total number of cannons. I advise trying this early in a character's career, or selling cannon until it is below 16.
Throughout the Caribbean are numerous settlements -- two of which will prove very useful for this achievement.
The two settlements of note are Indian or Pirate settlements. You can visit either and instigate an attack on a big city in the area. To determine the size of a city, large, medium, or small, go to the menu and look at the world map. If you zoom in to the middle of the map you should notice a building icon near some cities - if it is a large building, it qualifies as a large city, a small building indicates it is a small city, while a medium building indicates a medium city. A pirate ship or Indian war canoe will then leave the settlement and head to the city. Once it reaches the city, you will be notified, success or failure, by the sound of a sword clash followed by text in the bottom of the sailing screen describing the results of the attack. If the attack is a success, select the city, and you should see the number of cannon has decreased, which will increase your odds of a successful bombardment. If the message given when you select bombard doesn't indicate you will succeed, simply have the city attacked again until it does.
To ensure you have less than 16 cannons, I suggest using your starting ship as it carries 12.