This is very limited and it works in this contrived case because the match is exactly the literal alternation of the regex. However, the purpose of regex is to represent patterns and they are not the exact literal. In order to give it a change each regex must be tracked on its own and each line must be check times the amount of regex.

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In case that a visual aid is necessary.
example.file

Code:

abc1
Abc5
bcd1
abc2
abc3
abc4
Abc6
bcd2
bcd3
cde1
cde2
bcd4
cde3

You want three instance for each match. With the previously presented suggestion:

Code:

awk 'match ($0, PAT) && ++T[substr($0, RSTART, RLENGTH)]<4' PAT="[Aa]bc|bcd|cde" example.file | sort

Code:

abc1
abc2
abc3
Abc5
Abc6
bcd1
bcd2
bcd3
cde1
cde2
cde3

Instead, this could be a more realistic three matches of a pattern.

Code:

awk '/[Aa]bc/ && a++ < 3; /bcd/ && b++ < 3; /cde/ && c++ <3' example.file | sort

Code:

abc1
abc2
Abc5
bcd1
bcd2
bcd3
cde1
cde2
cde3