You have mentioned that boys tend to grow to a minimum of their parents' heights. My grandparents are on the shorter side, between about five foot and five foot five. Could it be possible that I inherited genes from them to make me shorter? Why do height predictions only rest on the parent's heights rather than the grandparents', if the grandparents are also responsible for contributing genes to the person?
It is because you don't inherit your grandparent genes directly. There are currently nearly 700 genes known to influence a person's height [Anthony Rivas, Height Isn't Only Determined by a Tall Gene, But by Hundreds of Genes; Some Not Even Directly Related to Growth].
Let's simplify the problem. Suppose a person inherits a two sets of genes, one from each parent and the result is that he grows to be the average of the two sets. The first set is what is inherited from the father and the second set is what is inherited from the mother.
Paternal Grandpa (5'8") (5'11") actual height: 5'9.5"
Paternal Grandma (5'2") (5'6") actual height: 5'4"
Maternal Grandpa (6'3") (5'11") actual height: 6'1"
Maternal Grandma (5'6") (5'8") actual height: 5'7"
Father (5'11") (5'2") actual height 5'6.5"
Mother (5'11") (5'6") actual height 5'8.5"
Child 1 (5'2") (5'11") actual height 5'6.5"
Child 2 (5'11") (5'6") actual height 5'8.5"
Unfortunately it is not as clean as this because so many genes are involved. Instead of inherited a specific height from a parent, you a blend of potential heights. But notice that even in this simple example, it is possible for children to be shorter than their grandparents. Basically, a child only has 1/4 of his grandparents' genes, but he has 1/2 of his parents' genes. It is the interaction of the specific set of genes that determines height.
Even with this, the current theories say that genetics control between 65% to 80% of your height. The rest is determined by environment, such as stress or how well you eat.