Absolutely, absolutely. I honestly think that a failure for kids to grasp the nuances of set theory is probably the one thing that inhibits them most. Notions like Zorn's lemma, axiom of choice, ordinal numbers, etc. are standard mathematical parlance yet are often breezed over by kids as "technical details".
I can not tell you how many times people have come to me with difficult problems in algebra or topology and I was able to answer them instantly. Why? I didn't have some topological/algebraic trick up my sleeve--I just noticed that the two spaces person X was trying to prove weren't isomorphic/homeomorphic were of different cardinalities! Never forget forgetful functors.
If I ever had the ability to institute one course any undergrad hoping to become a mathematician should take it would be "Category Theory AND SET THEORY for the Working Mathematician". Seriously, if you ever hope to get somewhere especially in point-set topology you better be damn comfortable with the prerequisite set theory.
EDIT: I would recommend reading the following series of notes by MSE frequenter (and fantastic expositor) Pete L. Clark, as these are as close to "Set Theory for the Working Mathematician" as you are going to get:
Finite, Countable, and Uncountable Sets
Order and Arithmetic of Cardinals
Arithmetic of Ordinals
Some Cardinality Questions
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