Some universities say that "The minimum number of letters of recommendation are 2 but we highly encourage 3". If I know that I can get 2 good letters of recommendation but they "highly encourage 3", should I send in a third?
They may count, or not, depending. My department would count because every application had to qualify for submission for graduate fellowships (policy decision), where 3 means 3. (There was also a quota of "exceptions", they could spend an exception if the candidate was worth it). Not all departments will care. Not all faculty and disciplines have fallen victim to praise-inflation, so a simple letter stating that you did a competent job in your work could be good enough, especially if the writer is known for being conservative in praise.
There are three different dimensions of the value of a recommendation letter that are relevant here:
The quality ascribed by the writer to the recommendee.
The certainty or itensity of that statement.
The reputation of the letter writer himself/herself.
A letter that scores high in the first category, but is less persuasive on the other ones, should not be detrimental. A typical example would be "CANDIDATE attend my course on X, participated well and scored top grades in the exam." This won't get you far, but also shouldn't hold you back.
A letter that is not good because of the first criterion is problematic. A letter which scores very low on the first and very high on the third will probably kill of an application regardless of most other circumenstances.
If only to clarify my comments to other answers: if the "instructions" say "minimum 2 letters, 3 strongly suggested", or similar, it means that the admissions committee reserves the right to consider situations with only 2 letters... but certainly not that it hardly matters. It matters. A solid 3rd letter is fine, and infinitely better than no third letter, for at least two reasons. One is the mere "strongly encouraged" message, which, if effectively disregarded, amounts to a "failed diagnostic". The other is the subtler issue about appraisal of a student's potential: as it happens, it seems possible to get one super-enthusiastic letter, and one more "pretty good", ... but it's hardly to bluff to a third that is "ok". Further, and perhaps surprisingly to students, the "second excellent letter" is very often not-so-sterling-after-all, and this unexpected loss can be compensated by hearing from a third, "disinterested but informed" party, that the student has good chops. So... everyone, including the student, wants that third letter.