Have you ever read that we only use 10% of our brains?

That's nonsense (just google "10% brain myth" for an explanation why). You've probably read it at the same time someone was trying to sell you something (or convert you to some religious/spiritual cause).

But it does beg the question... now that we are dealing with more information than ever, is there not the chance that we will eventually run out of space in our heads?

Or, as Homer Simpson puts it,

"Marge, every time I learn something new it pushes something old out of my brain"

Well, there isn't much need to worry. The brain does not work like a computer, with a limited number of bytes. Instead, it uses neurons and it is the connections between each neuron (synapses) that allows you to remember things.

There are over one hundred billion neurons in the brain, and each neuron can connect with thousands of others. Therefore, the potential number of paths signals can move around the brain is more than the number of atoms in the known universe.

Therefore, I don't think that we have too much to worry about in the near future.