Why is Hagrid not allowed to do magic?

Hagrid is Harry’s very first friend in the wizarding world. When he comes crashing into the hut-on-the-rock, armed with his pink umbrella, he changes Harry’s life forever. As the gamekeeper at Hogwarts, he is also able to look out for Harry during his time at school. 

As somebody who is clearly at home in the wizarding world and works for a magical school, you might be surprised to hear that Hagrid is forbidden from doing magic. He isn't even allowed to own a wand.

However, this wasn't always the case. Hagrid was a pupil at Hogwarts himself and was learning magic along with everyone else - until one horrible day in his third year, when he was expelled (for reasons we discover later on in the series). Although he was able to stay on as gamekeeper, his wand was snapped in half. So that was it, no more magic for Hagrid – or so it seemed.  

If you take a closer look at Hagrid’s pink umbrella, it does an awful lot more than just keeping him dry. After all, it does give Dudley that rather fetching pig’s tail. We also happen to know that Hagrid manages to keep the pieces of his broken wand. While he might deny still using them when asked by Ollivander, he does clutch his umbrella that little bit tighter. Curious…   

So, there we have it. Hagrid did have the ability to do magic, he just isn't supposed to use it after his expulsion from Hogwarts. Though strictly speaking, he has been known to bend - okay, break - the rules on the odd occasion, thanks to his trusty pink brolly.