There is a 2nd tone-2nd tone change in spoken Chinese, but it's different from what you described. Y.R. Chao's fantastic A Grammar of Spoken Chinese (unfortunately long out of print in English, though an abridged version is available in Chinese) describes a phenomenon where 2nd-tone words in the middle of trisyllabic compounds change to 1st tone. San Duanmu's The Phonology of Standard Chinese confirms this fact, but the author notes, "My own sense is that T2 Sandhi is not a productive or required rule. Instead, it is likely to be conditioned by the speed of speech (Shih 2005) and frequency of the expression."
Here are the details of this sound change, as presented in Duanwu's book:
In a trisyllabic expression, if (a) the first syllable is T1 or T2, (b) the middle syllable is T2, and (c) the final syllable is not weak, then the middle T2 can change to T1 in conversational speed.
(T1 and T2 stand for 1st and 2nd tone.)
I admit that I have never consciously "used" this rule myself while speaking Chinese, but, when I'm paying close attention, I do notice it arising in my speech and in the speech of others.
(There's no evidence in either of these books for the 2nd tone to 3rd tone change that you mentioned, so I assume that it's a unique nickname rather than a general rule.)
EDIT: I did a bit more reading in Duanwu, and was surprised to find that in Taiwan Chinese he notes widespread Tone 2 to Tone 3 shifts, but only on lexical boundaries. That still doesn't explain your situation, however.