Is it possible to break down clay soil with lime?
Some clay soils improve with the addition of lime - but it depends on the type of clay you have.
Far better to dig in as much organic matter as you can, well rotted farmyard manure, spent mushroom compost, homemade compost and leaf mould, as much as you can whenever you can for a few years. This will result in a really fertile and moisture retentive soil.
Dove is bang-on - it's always preferable to improve clay that way. In severe cases lime will help but it's best to do a soil test with a cheap kit first as adding lime to an already alkaline soil isn't a good idea - there are far more plants which prefer neutral to acid soil than those which prefer alkaline conditions. In really bad claggy alkaline clay, adding gypsum (builder's plaster is the cheapest way to buy it) will help it break down without increasing alkalinity. Frost will do a fine job, too - leave big clods of clay on the surface over Winter and they will be broken down a little every time there is a frost.
Clay is naturally very fertile and although it's hard work to improve the structure, once done you will have one of the best soils there is.
"Clay is naturally very fertile and although it's hard work to improve the structure, once done you will have one of the best soils there is.".
Just a reminder, though, it does take time to break down and improve and you won't do it in one go...I am only just seeing the first results after 18 months. I find that the best time to work anything in is immediately after rain when the soil will have softened. Then get as deep as possible and stuff as much organic matter in there as you possibly can. To DoveFromAbove's list I would add even the used compost from this year's containers as well as organic (uncooked) kitchen waste, it all helps to loosen the clay. No lime, though, the neutral to ever-so slightly acidic clay medium is the best for the widest range of plants.