As with many prescriptive "rules", the rule you mention (a) generally does not reflect the actual usage of educated native speakers, and (b) is based on spurious argumentation.
The argument is usually based on a couple of (probably largely unfounded) presumptions, such as:
- that "He's as tall as [me/I]" somehow has to be "short" for "He's as tall as [I am]"
- that sentences like "He's as tall as me" somehow introduce some ambiguity that can't be resolved by context/other natural means.
The spuriousness of the first suggestion can be shown, for example, if you look cross-linguistically. In French, "plus/aussi grand que *je" is not grammatical, and no French speaker is suggesting that the grammatical version "plus/aussi grand que [moi]" is short for "plus/aussi grand que [je (ne) suis]".
In the second case, did your teacher or does your grammar book actually point to a documented case where a real-life misunderstanding occurred because of "as/than me" being used where it "should" have been "as/than I"-- and if it does, does it explain why saying "as/than I", rather than "as/than I am" (etc) is the best solution?