Hmm, Forget-me-Nots is the name of several completely unrelated species of plant, all with similar blue flowers (plus some pink or white varieties). Perhaps you mean the small annual/biennial ones that grow to about 6" high, although they aren't bulbs. They are certainly hardy down to 10F or so, but you would have to introduce them to cool temperatures gradually. Plants grown outdoors should be hardy right through your winter.

Sorry - I should have been more specific - the species of forget-me-not is myosotis sylvatica. These were started from seed under lights - no they're not bulbs, but I'm planning on using them as a ground cover against my bulbs. (Have a bunch of daffodils, tulips) So I was hoping I could get these guys blooming at the same time as the tulips by starting them indoors a bit early. But I think I may have started them a bit too early anyway!

These are very hardy. Put them outside somewhere sheltered at first. Or put them out just during the afternoon for a few days, then for longer. You can always bring them in at night if it is forecast to be really cold. After a few weeks, you should consider them hardened off and then plant them out.

Is your soil thawed yet? I'm not sure how many flowers you'll get this spring, they tend to grow during late summer and autumn to flower in the spring. I have never tried growing them inside, maybe that will give them enough start to flower this May. They will self seed and you'll never get rid of them :) They flower in my garden with the latest tulips and for a whole afterwards.

I'm in Chicago, and I definitely wouldn't do it yet. Chicago has a pretty consistent history of temps at or below freezing until at least April 1, and frequently gets freezes right through May Day. They could possibly die of shock- either because they're so young, or because they may not be able to thrive in such a cold temp at that stage of maturity.

Better to transplant them into the biggest, deepest peat pots you can find, adjust your lights, maybe take them out on warm days, and don't harden them off until at least April 15 at the absolute earliest. April 21 would be even better. According to weather.com it looks like lows won't be consistently above 45 until then.

If you are really dead set on doing it, though, there are two plants that are called Forget-Me-Nots, from different genera - Cynoglossum and Myosotis. If what you have is a Cynoglossum, then they're far less hardy and you'd really be pushing it for the plant - it'd be far less likely to succeed. Myosotis is hardy to 3b.