You can ask for samples from current lots. That's what we've done. For stones that have a lot of variation though, you won't see all the possible variations but hopefully will get some idea how they will look.

Sometimes they look just like the sample, and other times they look like an entirely different stone. If you don't like what you've been given, refuse the order. There are also grades of certain stones, too, and you hiht on one of the requisites of determining the grade (of travertine, I'm assuming?). If it were me, I wouldn't accept it, from what you've said. Especially if you paid a premium price for it.

I didn't pay for anything yet; I haven't placed an order. These are the samples from current lots that I'm talking about. I'm glad I didn't forgo asking to see samples from the current supply. I don't like the samples I'm seeing.

I had the same problem! Had to basically re-select all bathroom tile twice because my sample tiles were so muddy and gray whereas the boards from which I had originally selected them were much whiter and much nicer. Ended up using natural stone (Bianco venatino) on bathroom floor but going with subway tile in shower bc I had already delayed my project for almost a month waiting for samples (which ended up looking completely diff than display boards). I also called every tile place in the DC metro area looking for in stock tiles so I could see the sample that day.

If it's a fairly common limestone, look for a larger dealer who might carry it open stock, that way you can look through several boxes and decide if it's what you want

As others have mentioned, look at the actual stock, not just the boards.
In fact, I like to buy from warehouses where they stock. I urge them to walk me through to the pallets and pick out the stone myself.
Here is a crazy idea but it is common in other countries.
Lets say you want to buy marble tiles. You negotiate a price (usually higher) where you get to choose each tile from the boxes they have.
If the store does it for you, it is called Select. Or you do the selection yourself.

Another strategy is to ask how much restocking cost is. Then buy double of what you need, keep the best pieces. Return the rest for the small penalty of restocking.

In other words, never go by board samples :)

Thanks for all the advice. Unfortunately I live in a small town so I don't have the opportunity to use some of your (great) ideas. I wonder if a 5 hour drive to a major metro area is worth it and how much tile I could fit in the back of an AWD station wagon? But the ideas of hand picking and of returning some are things to consider. Thanks.

When I was ready to buy some Black Galaxy granite tiles a year ago, I knew I was going to have to shop around for the primo stuff. The BG that I was seeing was lighter and didn't have near the gold and bronze flecks in it as it did 10 years ago. I asked the owner of a local stone yard to find the prettiest BG he could find. It took a couple weeks but he came thru for me! It was exactly what I was hoping for. No, actually it was better than I expected :) Maybe your local guy could call around to other stone yards or perhaps you could order samples from online stone vendors.

I decided that I needed to stop comparing it to the display board and just look at the sample tiles that we got from the currently available stock and decide if I liked them. I discarded 2 of the choices and like 2 other choices. We will check their availability tomorrow. I did drive 1.5 hours (each way) to the only place within 100 miles that had tile in stock. They didn't have anything better.

You have zero chance of the sample board being the same lot as the ones presently available.

Even manufactured tile varies lot to lot, let alone stone.

Look at samples from available lots.

I watched Sarah Richardson's show on HGTV last Sat. and she was using natural stone for a floor. She said you're going to get variations in color even within the same box to the point where they look like completely different stones. She advised buying extra and laying your stone tiles out before you start to set them.