Am I a Secondary Taxicab?

Jelly, 9 bytes

Credits to Erik the Outgolfer.

Œċ*3S€ċ>1

Try it online!

This is too slow that it won't even work for 1729 online.

Much faster, 12 bytes

Credits to Dennis.

R*3fRŒċS€ċ>1

Try it online!

Mathematica, 35 bytes

Count[#^3+#2^3&~Array~{#,#},#,2]>2&

Pure function taking a positive integer and returning True or False.

#^3+#2^3&~Array~{#,#} tabulates all sums of cubes of two integers between 1 and the input. (This would be much faster with a sensible bound on the integers to be cubed, like the cube root of the input; but that would take precious bytes. As it is, the code takes about 30 seconds on the input 13832 and scales at least quadratically in the input.) Count[...,#,2] counts how many times the input appears in this list at nest-level 2; if this number is greater than 2, then the input is a semi-taxicab number (greater than 2, rather than greater than 1, since a^3+b^3 and b^3+a^3 are being counted separately).

Mathematica, 38 37 bytes

Tr[1^PowersRepresentations[#,2,3]]>1&

-1 byte thanks to @GregMartin

As always, there is a Mathematica builtin to everything.