Onions really tiny?

I found my onions yesterday between all of my carrots. The leaves had shriveled to nothing, and the bulbs were half out of the ground. However, they're only about 2 inches across! I've never grown them before, so I'm not sure what I did wrong. I started them from sets in spring. Two things that came to mind:

1) Did I not plant them deep enough? I thought it was weird that they were out of the ground.

2) Did they not get enough light? The only things around them were lettuce in spring and carrots (right now). Both of those were started from seed, so the onions were taller than them until they started dying back. I made sure each had plenty of root space from the lettuce and carrots.

Varieties of onions grown in the north start to bulb up when the days start getting shorter (June 22). They store energy from the "leaves" to the bulbs. (They are biennials, and would shoot up flower stalks if you re-planted them next year.) How big your onions get depends on how big the tops get before June 22. And they need plenty of fertilizer, water, and light.

Having said that, I'd be happy with 2" onions! They are not the easiest things to grow.

As for their tops showing above ground - that's exactly the way they should be.