Yes indeed.

Which is why, Lisabelle, I recommend that you not cut.
Just hear me out. A strand of missing hair doesn't just grow back instantly. Even if you cut from a discreet place (or a few discreet places, as BB suggested), you'll still know that the short pieces are there. And once they start growing out, you will notice them. Especially when you try to wear your hair up. I notice mine most when applying leave-in's. The short pieces are now about 6" in length.

Now ask yourself this: If things don't work out with this new guy, how do you think you'll feel about having those short pieces left over after he is out of your life? Knowing that you've sacrificed part of your hair - that you've spent so much time and energy on - for someone who didn't deserve it?
I'm not trying to sound pessimistic - I'm thrilled that you've found someone that you feel close enough to to give away a part of yourself. But in the real world, these things don't always work out. In fact, they rarely do. So you must ask yourself whether you can live with the consequences to your hair if they don't.

With that in mind, let me suggest an alternative: harvesting.
Try collecting hairs as they fall out naturally and saving them all in one place. This is much more humane for your hair since you won't have to worry about any short pieces growing out. And it's also a better test for your guy since this process will probably take a few months. If he doesn't stick around for as long as it takes to collect a lock of hair, then he wasn't worth it anyway.

Don't get me wrong, I think it's terribly romantic to give a lock of hair to your lover. But for gosh sakes, girl, don't go hacking on yourself right away. Make him work for it! 
    I have done it both ways and that's why I'm recommending this. I don't want to see you do something that you'll come to regret. Believe me, I know all too well how easy it is to get swept up in the moment and want to act on those feelings now. But please consider the possible repercussions first. I don't want you to make the same mistake I did.