After doing some digging, it appears that this construction is rare but is attested in several classical authors.
Pinkster's Oxford Latin Syntax (p. 977ff) is my guide. He states:
The use of the genitives of the personal pronouns nostrum and vostrum instead of the corresponding possessive adjectives is rare.
Two examples from Cicero:
Recordamini qui dies nudius tertius decimus fuerit, quantus consensus vestrum, quanta virtus, quanta constantia. (Cic. Phil. 5.2)
This example "sounds" like a partitive genitive, but I still think the use of the genitive pronoun instead of the possessive adjective is remarkable.
Is enim splendor est vestrum ut eadem postulentur a vobis quae ab amplissimis civibus. (Cic. Att. 7.13.3)
This is an unmistakable example.
Above all, this construction seems to occur with the reflexive possessive pronoun, sui. Here are some examples:
Lysander Lacedaemonius magnam reliquit sui famam, magis felicitate quam virtute partam. (Nep. Lys. 1.1)
Non aliud Scaevolae Mucio cognomen dedit et capto contra Porsennam regem libertatem relquit quam vilitas sui. (Sen. Con. 8.4.1)
Pinkster provides more examples in op. cit. p. 978.
As for mei and tui, the only example of this usage is in the dubious context of a paraphrase by Priscian (who is decidedly post-classical). Glossing the Virgilian verse "Est mihi nata, viro gentis quam iungere nostrae...", he says:
nam ad scientes, esse natam, nomen autem proprium ignorantes, dixisset 'mea' vel 'mei nata, Lavinia est', subdistinctione posita post 'natam'.... (Pris. Inst XVIII)
As for nuances of meaning, the construction seems so rare that it is hard to speculate: in some of the above examples, hints of a partitive ("consensus vestrum") and objective ("famam sui") meaning appear. In others, the reason is less obvious.