I think the proper fit is to stand with your arm hanging straight down. The end of the sleeve should be in the middle of the palm of your hand. If you curl your fingers upward, you should be able to just touch the end of the sleeve. This may seem long, but when you move and bend your elbow or reach for anything, you need that extra length.

I reach forward and bend my elbows. If the end of the sleeve hits just at the wrist, it's a good fit.

Here's what my patient teacher (an Algerian tailor) taught me:

Fit with the customer's arms at his sides, having adjusted the garment to hang properly from the shoulders. Remember, for ANY tailored garment "fit" begins at the shoulders!

For a sport coat: the outside (placketed) side of the sleeve ought to fall 1-1/4" below the wrist bone. You may taper the sleeve hem up a scant 1/4" on the "thumb" side, or not. (I always do this, personally).

For a topcoat: the length of the sleeves should be judged to COVER the sleeves of the sportcoat/suit worn underneath it (I like to see 3/8-1/2" coverage)! If possible, have the customer put on a sportcoat and then mark the topcoat's sleeves accordingly. If the topcoat is worn without a sportcoat/suit underneath you may adjust the sleeve length up accordingly.

Given the shoulder pads and wadding in a constructed sportcoat are in the 1/2" range generally, I would add 3/4-1" to the guideline given for a sportcoat or suit jacket listed above.

Thanks so much for the responses!
After reading this, the sleeves are definitely too short, and I'm going back to the store to have them readjusted. Hopefully without a hassle!

I have never seen anyone in a coat with the sleeves covering anything but the wrist bone when the arms are hanging down. I live in the northwest--maybe this is a regional thing.

Just to follow up, I went back to the store and they are redoing the sleeves for me to lengthen them a bit, no problem and no hassles. I feel so much better now, thanks again!