Sounds like the finish was some sort of oil finish---tung/BLO/etc. that was too old to cure. Most oil finishes are used improperly anyway---they do not make a good finish for tables/chairs.

The splitting could have been caused by several factors---but probably happened due to the lack of a protective finish and cracked due to moisture variation cycles---drying/absorbing/drying with seasonal changes.

Use well(evenly) dried wood to make the table. When gluing boards together, reverse the grain pattern for each board. Attach the top to allow for seasonal movement. Use a good varnish and apply at least three coats to the entire piece, top, bottom, sides,edges.

Splitting can occur when the table's design doesn't allow for wood movement. All wood expands and contracts with changes in humidity, generally far more across the grain that with the grain. If the frame of the table didn't allow the wood to move, it could split. Some woods also are more prone to cracks and checking than others and those could have appeared post construction, especially if the wood was not sufficiently dry when the table was built.

I agree with Mac, some kind of old finish that wouldn't properly cure. Just make sure you buy a "fresh" can of whatever you use for your table.

Shellac that has been polished for years with alternate treatments of lemon oil (or scotts liquid gold) and wax will begin to soften.
Casey

And certain lacquers will soften and become sticky when cleaned with certain products.
As for the splitting...poorly designed and amde furniture.
Linda C

other conceivable causes:
- Shellac that was too old and had esterified and will never fully cure
- Finish that was exposed to acidic body oils and rarely cleaned causing the finish to permanently soften and turn to goo
- Build up of wax due to not removing enough when applying or apply too much. I have also seen this with too much furniture polish (like multiple times a day).
- Wax finish followed by emulsion-based polishes that soften the wax
- Pledge, the evil in a can

As far as the splits go,
- cross-grain construction on too-wet wood to begin with.
- bad glue jobs
- excessive heat and/or humidity exposure (e.g., attic or basement)

One other thought: people used to (still do?) spiff-up a piece of old furniture when it became dowdy; if the old finish was already dirty/oily/degraded, the new finish will fail, sooner or later, and sooner can include never fully curing. So, a soft finish can be from a fresh coat of some varnish over a dirty ol' finish.
Casey

Try wiping down with paint thinner.

If the problem is wax or oil this will remove it without damaging any finish (including shellac).