Can a 96 percentile in the CAT get admitted in XLRI or any IIMs?

For XLRI, you will have to appear in XAT. As for IIMs, the selection for admission varies depending on which category you belong to.

• General Category Male Engineers (GEM): You are most likely to get calls from the new IIMs. It is more or less the same for GEF except for the gender diversity points. 
• General Category Non-Engineers: If you are from Arts background, expect a call from IIM A, B, C, even if you have only above 80% in 10/12/Undergrad. If you are a Commerce student you can expect a call from IIM I/K/S apart from the new IIMs.
• OBC/NCL Category: As the sectional cutoff is lower, you can expect a call from all IIMs except ABC if you are an engineer. Female and non-engineering students will have gender/academic diversity advantages.
• SC/ST Category: The required cutoff is fairly low for these two categories. But you will need decent past academics (at least 80%) to be eligible for top IIMs.

But these are only the minimum requirements to get a call for WAT-PI at these institutes. To be selected for admission, you will need to perform well in the WAT-PI round too as the competition will be fierce.