In Changeling (2008), Cops handed over a fake kid to Collins, then pretend for it to be the real kid. Then why can't she demand a DNA testing on this fake kid?
The film is based in 1928 and the DNA profiling as we know it today was developed in the 1980's, with the first instance of it being used legally in 1987.
In 1987, Florida rapist Tommie Lee Andrews became the first person in the U.S. to be convicted as a result of DNA evidence; he was sentenced to 22 years behind bars
TL;DR : The technology was not around at the time.