If I talk about JEE Advanced - the questions do not test a very deep knowledge of mechanisms. The questions are usually framed by combing many elementary reactions together and asking about the product or combing it with other chapters like stoichiometry. Named Reactions(only those in syllabus like Aldol,Cannizaro,Perkin,Hoffmann Bromamide) are very important for JEE Advanced and you should know every aspect of these reactions including their kinetics(order, molecularity and how these change with concentration of substrate). Also for Advanced stick to the syllabus.
However sometimes in JEE Mains you can find some tough mechanism problems with confusing options where some extra knowledge of mechanisms can help.