For the available solutions, it may help to understand how acrylic paints work.
Acrylic paint composition
The main ingredients in acrylic paint are pigment (colored powder), and acrylic binder. There may be additional ingredients to control drying time or characteristics of the uncured or cured paint, and cheap paints may mix in other, less expensive binders or cut costs in other ways. But the gist of the issue relates to the acrylic binder.
Once the binder hardens, it creates a film that is essentially Plexiglass that holds the pigment particles together, and (if the hardening happens after you have painted with it), holds them to the surface. It isn't "pure" Plexiglass; it's adulterated with other stuff and the polymerization may not be complete. The film is microscopically thin layers of plastic around the pigment particles that is fragile compared to a sheet of pure Plexiglass. So almost-cured paint can be broken down or often partially resoftened. But the cured binder within it is the same plastic as Plexiglass, and isn't affected by water or dissolved by acrylic medium.
The pigments are basically inert. All of the action relates to the binder. The paint starts with a certain ratio of binder to pigment. When the binder hardens, that will leave a certain amount of plastic between the pigment particles, which affects opacity and strength.
Thinning acrylic paint
If you thin the (uncured) paint with acrylic medium (just the binder with no pigment), the paint will be less viscous because the pigment acts as a thickener. You also get a higher ratio of plastic to pigment, which will be stronger but less opaque. If you thin the paint with water, you don't change the ratio of plastic to pigment, just the viscosity (which also affects the thickness of the paint layer and its opacity).
How acrylic paint hardens
The acrylic binder hardens through polymerization. In its liquid state, the binder contains the precursors for the plastic, which are small molecules, suspended in water. There is also a catalyst that causes the small molecules to link together into long chains that become the plastic. The water keeps everything separated and the catalyst very diluted. When the water evaporates, polymerization starts. Some of the thickening of the paint is due to water evaporation (higher ratio of solids to liquid), and some is due to the start of formation of the chains of plastic.
Polymerization isn't reversible, at least not in a way that retains the characteristics of acrylic paint. If you thin the thickened paint, any of the molecular chains of plastic that have already formed remain, they just get spread around in more liquid.
As long as all of the solids in the paint (aggregations of pigment particles and plastic chains) are still microscopic, you can get the paint to still behave like paint. If you thin with acrylic medium, the new medium can bind together the previously polymerized plastic. If you thin with water, you slow down the polymerization of more of the existing binder. However, already polymerized plastic will be floating around in the diluted remaining binder. When the aggregates of solids get too large or the proportion of uncured binder too low, it will no longer act like paint.
Restoring paint that has only thickened
As long as the paint has not yet hardened, you can generally thin it. If you use water, use very warm, but not hot, water. With either water or acrylic medium, allow a lot of sitting time and do a lot of mixing to completely diffuse the liquid evenly throughout the paint. Once the paint has thickened, the catalyst has become more concentrated and some amount of polymerization has started. If you thin it unevenly and leave clumps of thickened paint, those areas will continue to polymerize even though the paint seems thinner over all. So it will have a much shorter shelf life.
Restoring paint that has dried on the surface or contains dried chunks
The dried paint won't be restorable, but it may be a skin covering paint that is still good. Try to pick out the solid pieces. Add a little warm water (drops) to the solid pieces and let it soak for 10 or 15 minutes. If the paint softens, chop and crush the chunks to expose any liquid paint. If that paint is still overly thick, add a little more water or acrylic medium, mix it in and let it sit again. Pick out bits or chunks that don't dissolve.
Paint that is hard throughout
This has already polymerized and can't be turned back into normal acrylic paint. However, you can sometimes turn it into another type of paint. If you use a solvent to dissolve the acrylic, you can make a paint that will re-harden when the solvent evaporates. It probably isn't worth it, though. Here's why.
The process and resulting paint are hazardous. There are solvents that will dissolve acrylic. You can buy small containers of acrylic welding solvent (usually mostly methylene chloride), but that is designed to evaporate extremely fast (probably too fast to be good for paint), and you don't want to breathe the fumes while using the paint. Readily available ones often found in a paint department include toluene, acetone, MEK, and probably xylene or lacquer thinner. Some people report using nail polish remover (ethyl acetate).
Most of these solvents are highly flammable, hazardous to breathe, and not good for your skin. So you need ventilation and protection. Consider whether you want to do your painting wearing a respirator.
The result usually doesn't make good paint. To turn it back into paint, you need to carefully control the ratio of solvent to dried paint. It can take days or even weeks with periodic mixing for all of the acrylic to dissolve and create a smooth, uniform paint consistency. It is hard to control the consistency. Depending on the solvent, the mixture may not handle well as paint (e.g., it may skin over quickly), and it may quickly harden or gel in the container, so it won't have much shelf life. The new mixture may not bond well. It may be brittle and will not have the characteristics associated with acrylic paint.