Overcrowding (as Chelle mentioned), heavy soil, and insufficient/uneven watering can all cause radishes not to form good roots. You're right that too much nitrogen in comparison to other nutrients (primarily phosphorous) can encourage leafy growth as well.
For your next attempt, try working some compost into the beds a few weeks before you plant, then mulching once the plant tops are large enough not to be covered. Those two things will help with soil texture, water retention, and nutrient balance... the spacing of the plants you'll still need to take care of, either by seeding further apart or thinning aggressively.
