The veneer tore because the substrate had a lot of movement, either because of a shock, or because it was a bad design. I recommend you pass on it, unless you like a tablecloth.
Casey

Casey, I think you are spot on about a structural cause because the cracks are mirror images on that side of the table. However, let me ask this- can I strip, fill, sand, stain. then wax that tabletop a darker color- would that make this crack less apparent exp if I try to smooth out that area? The top seems lighter a few shades than the pedestal legs and needs to be darkened anyhow. I really like the shape of this table and chairs.

Like Casey, I recommend you pass on this one. The table was poorly made to begin with, in ways that are essentially impossible to fix. It looks as if they edged the top with wide pieces of solid wood before veneering. That's a bad idea to begin with, but with that much movement I'd guess the solid wood they used wasn't properly dried either. The veneer is not only cracked but also delaminating in places (notice the bubbles between the crack and the table edge). Anything you do to mask the damage is going to be a temporary fix; this is likely to keep getting worse.

Er... I could be wrong about the bubbles. Maybe those are just reflections of the light fixture. Even so, this table isn't worth futzing with unless it's so cheap that you're willing to live with it as-is.

"table veneer has developed cracks in two places"

As others have noted, that is a lot worse than a problem with the veneer.

The substrate under the veneer has failed, and nothing you can do to the veneer is likely to have any affect on the crack in the substrate.

you would probbaly have to remove the veneer completely (both sides), repair the substrate (if it can even be repaired) and then replace the veneer.

I would almost bet that the substrate is some type of particle board that has broken, making a repair nearly impossible.

Pass on the table unless you are willing to replace the entire top.

The reason veneer is used often is because a very small amount of wood can be used to create an inexpensive piece that would have been very expensive or impractical is solid wood were used.

The down side to that is that very beautiful pieces can be made that are actually very cheaply made. And a cheaply made piece is generally not worth fixing due to the early trouble and the fact problems will undoubtedly reoccur.

Restaining is a problem, since you have to remove all the existing finish. That means stripping or sanding. Stripping will cause the chemicals to seep into the cracks---and you will not be able to get it all out. Not good at all.

Sanding veneer is a plan of failure somewhere. I've yet to do/see/hear of a successful attempt to sand finish off factory veneers.

So, even if you were willing to spend the time and effort, there is a very small percentage of success.

Just for example, I tried the same thing on a dining room table twice. Once before I had much experience with or understanding of veneers. Failed miserably.

Then a couple years ago, I had an opportunity to try again for a relative. I have much more experience, having applied veneers, and a much more detailed knowledge of veneers. Plus I have much better tools and a LOT of experience sanding.

The failure was almost as spectacular as the first one because the second veneer was MUCH thinner.

OK, I hear you guys! No go on the table. Thank you for the strong advice and words of experience and wisdom
:)
Nancy