In my opinion, there are different levels of laboratories. For example, labs that LE depend on are held to a very high standard and significant backlogs are not unusual. Yes, I believe certain cases can be rushed, depending on circumstances.
A local lab that handles DNA for, say paternity/custody issues, are held to a certain standard, of course, which IMO may not be as high as the LE/Fed crime lab. These paternity/custody results, while important, are not as critical as a criminal case where someone's freedom might be at stake.
There are other, private labs where DNA can be done for -- I hate to use the word 'fun' - lesser important purposes...i.e ethnicity (how Irish are you?) or entertainment purposes. I imagine Maury might use this expedited option.
I'm not suggesting that any of these labs are shady or cut corners. Each may serve their purpose. A guy who thinks he may be the father of a newborn may not need the level of a criminal lab. If he wants to dispute the results, he can elevate it further....JMHO.