If debt is cheaper than equity, do companies always try to maximize debt? Why or why not?

We learned in class that debt is cheaper than equity, and that it increases enterprise value because of the tax shields. While I understand that a company cannot use 100% debt (otherwise no one would own it), do companies typically try to maximize debt and have as little equity as possible? Why or why not? Is there such a thing as too much debt? Of course you have to be able to pay it back, but don't you also owe something to equity holders?