Once samples from all sources are in hand, analysts isolate a bit of DNA from each sample, make lots of copies of it, and then process the copies through a machine that analyzes genetic markers -- DNA fingerprints -- that have been passed down through a subject's family. Typically, Bieber said, DNA tests examine around 15 of these markers. ... It typically takes several hours to complete each step of the analysis process, Bieber said, though he noted that in high-profile cases like this one, law enforcement agencies might already have genetic profiles of the relatives available -- which means they’d only have to complete one additional test.
I checked and it seems that there are paternity testing outfits that offer one day turnaround. So I guess it's not implausible that they could have pulled this off. I assume they still use variable number tandem repeats for DNA profiling? Update:Yeah, short tandem repeats.