J, ^{32 26} 23 bytes

24 60 60*/ .>(3$100)&#:

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-9 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, taking only the last 3 digits (this is how mixed base #: works), check that all of them are within the allowed range.

J, ^{32 26 23} 16 bytes

60*/ .>100#.inv]

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-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.

J, ³² 26 bytes

24 60 60*/ .>_3{.100&#.inv

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-6 bytes thanks to Adam

Convert the input to base 100, take the last 3 digits, check that all of them are within the allowed range.

J, ^{32 26} 23 bytes

24 60 60*/ .>(3$100)&#:

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-9 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, taking only the last 3 digits (this is how mixed base #: works), check that all of them are within the allowed range.

J, 32 bytes

[:*/24 60 60>_2".\_6{.'00000',":

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Prepend '00000' to the stringified int, take the last 6 _6{., format them in groups of 2 _2".\, and make sure all chunks are in the correct range.

J, ³² 26 bytes

24 60 60*/ .>_3{.100&#.inv

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-6 bytes thanks to Adam

Convert the input to base 100, take the last 3 digits, check that all of them are within the allowed range.