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Infants can only perceive in one dimension (closeness and farness). Young children can only perceive three dimensions (they can't plan or handle time). Teenagers and young adults can perceive 3+1 dimensions. However, this is all psychological and dependent on our personal growth, which means that someone could exist who perceives more dimensions than 3+1 and it wouldn't be a logical contradiction.

Is there any way to empirically test whether there are more dimensions? If there were 3+2 dimensions, then I could have a clock that is two-dimensional. Or I could have one clock that would tick in one dimension and another clock tick in another dimension.

How can we design an experiment to test this?

ArtIntoNihonjin.
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I don't believe that it's generally possible to test this in-principle. Assuming that we are part of a multi-dimensional manifold but that all our physics is limited to a 4-dimensional sheet within it, there would be no experiment that will show that there are more dimensions.

Now it could be that not ALL our physics is limited to this 4-dimensional sheet. In which case experiments in our 4-dimensional sheet could interact with bodies that enter and exit from other dimensions, and thus probe these dimensions. There are experiments in the LHC that may probe such eventualities.

https://futurism.com/investigating-higher-dimensions-with-the-lhc/

PhysicsTeacher
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