Carrots, turnips and onions? Barley and flax? Juten puffs and mage caps? Only carrots, turnips and onions can grow in multiple biomes. Barley and flax only in the Plains and the Mistlands mushrooms only in the Mistlands. The other thing may be overhanging shade that the wind may have blown out of the way to show "healthy" but when the wind dies down covers the plant and denies it sun light and good health.

crops die if they
- have a roof/ building(wall??) very close
- get eaten (basically, mobs attack crops and kill them, and animals like boar are attracted to do so)
- are too close to each other (can say healthy for a bit before saying needs more space, if its really borderline)
- wrong biome (can say healthy for a short time before getting sick)

to counter this, generally,
loft a farm up 5 tiles or so away from your house/buildings (this isnt much, like 1 floor tile gap)
plant every other square using 2x2 floor tiles as a guide
plant in the right soil as above ^^

this may be excessive, but it will work.

To add to the above if you hit the left mouse button you can attack and destroy healthy carrots, onions, and turnips. If you attack barley or flax it just harvests it if its ready.

Also note that if you have planted long rows and are harvesting that row in a continuous action after a certain number of consecutive pickings the sound will cut out and the picked crop icon will not drop to the ground or auto gather (this is a bug) and one or more plants, if you continue picking crops, will disappear. If you stop picking when the sound cuts out and let the program catch up you can prevent this bug from affecting you.