The animators done goofed.

As someone who's studying animation (and writes his own comic series) I can tell you that keeping proportions consistent is very hard to do. In animation it's even harder since you have to draw the same character(s) so many times. Comic artists thankfully have more leeway in that regard.

I haven't seen American Tail since I was real little so I don't remember too much about it but it seems pretty strange to me that a rat character would end up being as big as cats, especially since the cats in American Tail are supposed to be anthropomorphic. It could be done to make the character seem more intimidating. Eiichiro Oda, the creator of One Piece does this often with his villains. He'll make them look much bigger in certain panels to make them appear more threatening.

Likely not. As noted before, it can be hard to keep the porportions of a character the same. It's not as easy as you might think it is.

It would have been hard to re-animate, especially during the time that this was made. There was a chance that they noticed, but were unable to go back because it would have required an extra ton of work. Animation is a painfully long progress, and to fix the size of one character would call for reanimating other parts as well, racking up the amount of time needed to finish it.

I've seen that sort of thing a lot in some cartoons.  I just always put it off to it's hard for animators to keep the exact proportions the same between different characters and objects.  Like in the 70's Godzilla cartoon series he kept changing size all the time.

I asked this question once again on the Don Bluth Films Facebook page, and this was their response:

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Good catch. You are absolutely correct. Yes, it was "sort of" intentional. And no, there were no potions involved.
  Warren T is a runt cat, smaller than the cats but larger than the adult mice, not by much. The idea was that he was the bully of the cats. A little guy with a huge ego and a bullying attitude, sometimes referred to as a Napolean complex. And, with his small size and a crude disguise, he could pass himself off as a rat and a "friend", offering to help new-comers to America, get a job. Tiger on the other hand was a huge cat, two to three times as big as the cat gang members, and literally towered over Fievel.
Many times, trying to stage the various character sizes, like Warren T. with Fievel vs, Papa with Fievel, together within their environment meant we took some liberties or artistic license to avoid cutting off parts of their image on screen and give the shot a believable and pleasant appearance, without having to draw tiny minute images of Fievel.
Actually, as far back as I can remember, we've never had a viewer write to us about the discrepancy in the Warren T's size. We thought that some in the audience might have brought up the size changes on Tiger during the "A Duo" song performance. Which, was a much more radical discrepancy. But, no one did.
  Again, good catch and thank you for asking about it.

Hmm, interesting insight on Mr. Bluth’s part. Yes...

I only saw this film several months ago, nearly a year, and it was so... so incredibly hard on Fievel for him to go through all of those things, and still find his family while meeting new friends blew my mind for me. It was just overwhelming.