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... for an individual practicing law to specialize in a few completely different types of law. For example, would you ever find a person who is a specialist in corporate takeovers as well as some aspect of patents? And if those people do exist, is it possible for them to be high quality/high demand in one or both fields?

It is very unlikely that someone would do both IP and M&A work. It's much more likely that there will be some crossover between two related departments. For example, a lawyer who normally does M&A work might step in and work on financial restructuring (i.e. bankruptcy) deals when things look down.

That's biglaw. In a smaller firm, it's much more likely that the practice will be a bit of a hodge-podge.