Easy to handle..
What type of Nectarine is it?
What type of rootstock is it on?
Whats your irrigation and fertilizer schedule

and how did you prune it last season?

any information on a spray program will help too..

Turtleman, the trouble is, I have no idea what kind of nectarine or rootstock, since the previous owner of the property didn't know. It appears to be a dwarf variety, about 10 feet tall. The fruit had a lot of red coloration in the flesh, especially close to the pit. I was watering it twice a week during the hottest part of the summer, moving back to once a week now. I pruned it very lightly last season; it has a good, open vase shape. I believe I need to prune it more heavily this winter, as well as picking off more fruit when it first sets, there were too many per branch.

So what do you think? Thanks!

Rootstock/Type are important but I think your on the right track for pruning/thinning. Adding a good fertilizer in the Spring will help (slow release (3 or 6 month)) with at least a NPK of 15-15-15, your P-K will give you your blossom and set. I irrigate/flood our trees once every 10 days but we deep water and mulch the soils in Marana are a bit faster there than I have here so just make sure you dont over water.
The one thing about nectarines is the fruit won't size and/or scare if you've had problems with thrips, carry out a spray program

Check to make sure you see the bud graft.

It might have been replaced by a sucker from the rootstock during a bad pruning session. The sour orange rootstock produces small, easy to peel, sour fruits.

Ummm... Err,,

You cant grow stone fruits on citrus root stock

Thank you Turtleman. I did fertilize last spring, but I did not spray, and I know I should. I will buy some horticultural oil and a sprayer this year.

Any tips on keeping the birds off the fruit? I tried covering the tree with bird netting, and it kept blowing off, and was a tremendous pain to get over the tree. Very frustrating. I'm wondering if I can bag the fruit, like they do for grapes.

Holographic tape works well just remember to leave a long tail on it so it moves well in any wind and then change its location each 20 to 30 days or so, that way the birds wont get "use" to it, works well in the nursery here, theres also "Bird-a-away" as a spray. A little more expensive item is Ultrasound, but its a one time cost and preforms very well.. there the items I'm going to install in the nursery here this next season I hope

DUH! I was thinking TANGERINE :(

/me slithers back under her rock.