J, ^{32 26 23} 16 bytes

60*/ .>100#.inv]

Try it online!

-16 bytes (!!) thanks to Adam. This new solution uses the approach from his APL answer so be sure to upvote that.

Convert the input to base 100, check that all digits are less than 60.

Note the most significant digit is guaranteed to be less than 24 by the allowed inputs.