IMHO/E, yes. It isn't always easy to do (it rarely is for most), but it is certainly possible. I have noticed it in tanks that mainly house "sps" type corals, which show signs of deprivation much faster than other corals (other than maybe many soft corals). It basically results in a "failure to thrive" sort of situation where there is little to no algae, pristine water parameters, and very poor looking corals.
That being said, I do still advocate good skimming and I do tend to oversize my skimmers. Just don't go overboard
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Actually, it all has less to do with "trace elements" than macroelements, like nitrate, phosphate, etc. You can limit those to enough of an extent to cause problems. Most trace elements haven't been proven to be useful in one way or another short of anecdotally. The only things you should worry about is calcium, alkalinity, and magnesium--and be sure to test for those.