See BOL:
CREATE INDEX, DROP_EXISTING
"Because nonclustered indexes contain the clustering keys, the nonclustered indexes must be rebuilt when a clustered index is dropped. If a clustered index is recreated, the nonclustered indexes must be rebuilt to take the new set of keys into account"
I learnt this the hard way...
Umm, whatever you learned the hard way, I doubt it was due to DROP_EXISTING. What BOL is saying is that it rebuilds NCIs if need be, otherwise doesn't waste the time. However I see one caveat "If a clustered index is rebuilt to a different filegroup or partition scheme, the nonclustered indexes are not moved to coincide with the new location of the clustered index." Not something I'm ever likely to do, but figured I'd mention it, since I was about to claim that DROP_EXISTING never gets you in trouble. Anyway, unless you're doing something advanced, in a complex setup, it won't.