Since we are entering the winter (in the northern hemisphere) many people try to daven mincha at mincha gedolah. I have been told that waiting for a half hour after mid-day is a way of ensuring that one does not accidentally daven before mid-day and there are those who state that someone who makes a mistake and davens within that time period does not have to repeat Mincha.
Note that "a half hour" is actually shaos zmaniyos rather than 30 clock minutes and mid-day (chatzos) is calculated based on sunrise and sunset.
If one prayed prior to six and a half hours, according to many one has not fulfilled one’s obligation, however, some say that one has fulfilled one's obligation after the fact. 
 Mishna Brurah 233:2 writes that this is implied from Shulchan Aruch 233:1 (and is agreed upon by many achronim), however there are some achronim who say that after the fact one has fulfilled one’s obligation. Kaf HaChaim 233:2 quotes the dispute of the achronim but concludes that because Tefillah is derabbanan one does not have to pray again.
Does the time specification start at the very beginning of Mincha or as of Shmone Esrai. The difference would be starting Ashrei. While this is not really a time difference during the week, Shabbos Mincha could be as much as 10 minutes earlier. Similarly this could apply to a Ta'anis Tzibur since we must read the Torah.
The answer to this question could also make a difference according to those who say one is not Yotzei bed'avad.