I haven't officially been diagnosed with Graves or Hashimoto's but I seem to have 'gestational' thyroid issues.
Am currently taking meds whilst pregnant (35 w)
and have been since 6 weeks ish.
I had 2 m/c (blighted ovums) and it was explained these could be due to my thyroid.
Never had history of issues before, regular 33 day cycle, never had symptoms that I thought something was up.
BUT only discovered between the first two pregnancies when my TSH levels were nuts.
I was told if not treated could lead to more m/c.
BUT again this bubs snuck in, I wasn't on meds as was waiting for TSH levels to change naturally.
Weird thing for me is that previous two bubs didn't develop, yet this one less gestation, developed while my levels were low!
Straight away was out on thyroxine.
Endo advised to stay on it until at least 3 mths after bubs, as thyroid can be a contributing factor to post natal depression, and if we plan for another bubs within a 2 year window, Enfo suggests to stay on it.

Yes it's possible to get pregnant, patience, don't be harsh on yourself, try for stress free environment, and if need be press for other tests. I have heard of some with Graves/Hasimotos having celiac's, and other dietary issues.