Are there any situations in which L'Hopital's Rule WILL NOT work? Today was my first day learning L'Hopital's Rule, and I was wondering if there are any situations in which you cannot use this rule, with the exception of when a limit is determinable. 

There are situations where all conditions are met, but the usage of L'Hôpital leads to circular reasoning. For example,
$$\lim_{x \to 0}\frac{\sin x}x = \{\text{L'Hôpital}\}=\lim_{x \to 0}\frac{\cos x}1=\cos 0=1$$
Here, you have used the fact that $\frac d{dx}\sin x = \cos x$, which is proven by using $\lim_{x \to 0}\frac{\sin x}x$, which can be found is $1$ by use of L'Hôpital, which uses the fact that $\frac d{dx}\sin x = \cos x$, which... 
More information can be found in the not so meta-ish, but still excellent, meta.math thread on the subject here.
