Per comments, OP's instructors are probably just incompetent (or perhaps he somehow misunderstood what they meant). In standard English, and has two distinct meanings...
1: Dick liked Jane and she liked him
2: Dick insulted Jane and she slapped him
...where #1 simply uses and to join two "independent" statements - stripped of pronoun substitutions,...
1a: Dick liked Jane
and (another statement, with no specific relationship to the first)
1b: Jane liked Dick
But in #2 the conjunction implies a temporal/causal relationship between the two statements...
2a: Dick insulted Jane
and (subsequently - pragmatically, probably consequently)
2b: Jane slapped Dick
Grammatically/semantically, the and in #2 can be replaced by and then or simply then. Perhaps OP's instructors felt he was overusing the two-word form in contexts where it was pragmatically obvious that the second event was later in time and/or caused by the preceding event. But this is a matter of style, nothing to do with grammaticality.
TL;DR: Even if OP's instructors knew what they were talking about, they've obviously failed to convey the correct information to their student. The only reasons to avoid and then are (1) - if the context implies no [con]sequential relationship, and (2) - because using it excessively can result in a stilted prose style.
EDIT: In fairness to OP's instructors, they might be (mistakenly, imho) attempting to impose notions derived from prescriptive grammar. Consider...
2a: Dick insulted Jane
2c: Then Jane slapped Dick
2d: Jane then slapped Dick
2e: Jane slapped Dick then
...where 2c-2e are all valid sentences/clauses that could follow 2a. It's irrelevant here whether we put a period or a semicolon between 2a and whichever of 2c-2e we choose to use (you wouldn't hear a difference in speech anyway). The point is the fact that then can be moved around like this implies it's an adverb, not a coordinating conjunction. As that link says...
Be careful of the words then and now; neither is a "coordinating conjunction", so what we say about coordinating conjunctions' roles in a sentence and punctuation does not apply to those two words.
I cant say I find the above caveat particularly enlightening, but it may help explain some of the antipathy directed at the usage and then.