Well I'm 23 and he's 25, I know there's false positives and I wouldn't do anything regardless. He's adopted and we have no information on his family history.. so I really don't know.
I didn't with all three of my pregnancies. I wouldn't terminate if there was an issue so why go through all the what ifs and guessing. If there is a family history like heathert_kc mentioned then I would definitely consider it.
I decided to forego those types of testing. I knew I did not want to do any invasive testing because they come with a risk for miscarriage, though low. I would have been okay with doing non-invasive testing such as a triple or quad screen, but because I work for a Catholic organization, though I am not Catholic, I learned that these (like birth control) would not be covered. Plus I learned at 8 wks I was having twins which means the already questionable effectiness of such a test would be even lower. Plus I have to get ultrasounds every 4 weeks because it is a multiple pregnancy.
I think the most important questions to think about or discuss with your partner is what is your risk? Do you have a personal or family history of babies with genetic problems? Are you guys of advanced parental age (>35 for you or >40 for your partner) And ultimately what would you do if you learned there was a 'problem'?
The last one is what made my decision for me- I knew that I did not have the heart to abort, no matter what the test showed.