Special thickness veneer is still available. 1/42" was the standard a few years ago. I'm not sure what export specs are today. Thinner veneer is possible not only because of the ability to slice that thin but also because manufacturers have the machinery to apply it and sand without sand through.
I have pressed 1/32" veneer and it is a treat. It is less apt to overlap in layup, you can edge glue it for stable joints and sanding is less stressful for me.
1/16" on the other hand is too stiff and prone to splinters. I have seen 1/24" advertised and would be the max thickness I would want to work with.

Machines to cut veneer were developed prior to 1850, both sliced and sawn. The American Empire period, around 1820, is when you start to see more veneered surfaces and maybe lower quality building practices based on cheap labor. The veneer was/is thicker on those pieces but not 1/16". What the actual thickness was will vary. Would thicker veneer make it a better product? maybe. I do like to use thicker veneer but the full 1/16" can be to thick at times for some applications.

Some of the veneers sliced today are just to thin. Hammer veneering, if you even do that type of work, is that much harder with thin veneers.

Hide glue is not a bad glue at all. Its just not the right glue for certain situations.

I appreciate the above answers, but can a ¼ inch American hardwood of nice design be used as a veneer, or would it have to be less than a ¼, if so how much less?

The reason I am asking is because I have good looking hardwood pieces that are a ¼ inch thick, but they could be planed thinner if recommended.

Thank you.

You are kicking at a hornet's nest. I do not consider 1/4" wood veneer. The reason veneer works is that the expansion forces of the veneer are kept under control by virtue of the small cross sectional area compared to the substrate. Once the veneer thickness is greater than 1/8" then the veneer starts to exert forces that make cross grain gluing or wide surfaces problematic.
There are furniture makers that saw thick 'veneers' and use them in pieces but they tend to be small scale and often skins on a floating panel.
I can't understand why someone would want to resaw his or her own veneers after looking at the selection available from numerous suppliers.

There are rules, then there are rules. So, if 1/4" is to be thought of as solid wood, then what about these wooden building panels?

3 or 5 or 7 ply, with 2x materials, These panels should soon exhibit self-dectruction by all the logic we have amassed here at Woodweb. Multi-storey buildings are being built in some parts of the world with these. Yet they do not self-destruct - Why?

Plywood. The cross grain movement of the wood can be dramatic. Splitting off stone in a quarry, for example. It is hard to deny. Yet some folks do deny those cross grain properties and go right ahead making these panels.

Now I see them everywhere as 3 ply 8" x 10" panels used by utility companies to provide a clean, clear deck so their heavy equipment can work around without churning up mud. The panels arrive in a stack about 6' tall, and the three-ply panels are lifted of by a fork lift and distributed around.

Cross Laminated Timber