A multimodal likelihood function can have two modes of exactly the same value. In this case, the MLE may not be unique as there may two possible estimators that can be constructed by using the equation $\partial l(\theta; x) /\partial \theta = 0$.
Example of such a likelihood from Wikipedia:
Here, see that there's no unique value of $\theta$ that maximises the likelihood. The Wikipedia link also gives some conditions on the existence of unique and consistent MLEs although, I believe there are more (a more comprehensive literature search would guide you well).
Edit: This link about MLEs, which I believe are lecture notes from Cambridge, lists a few more regularity conditions for the MLE to exist.
You can find examples of inconsistent ML estimators in this CV question.