2 added 1103 characters in body
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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.

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.

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.

1
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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.