My dad's 5'11 but his mom's 5'4 and his dad's 5'5. Is that normal?

It's all in the genetics. You may inherit his genetic code for tall height, and you may not. These things are hard to tell just from looking on the outside. If any of your ancestors had the "height gene" for tallness, then you're a possible candidate. However, the same goes for the "short gene." The dominant gene will in most cases take over. Take a good look at photos of your relatives, and try to find the oldest pictures of your family. If you notice most of them as tall, then the chances are that you'll be tall as well. On the same note, average height and shortness constantly showing up will mean they're your best bets.