Suv Wheels

My stock rims Ford Explorer are designed for a 235/70/16 tire. Would it be possible to put bigger tires on the same wheels I have now?

You can put more / wider tires on rims same. I went a couple sizes on my without any problems. If you're overweight, you have a problem with the port from the center of the tire and you also have to watch and make sure you're not going to rub a fender or the firewall when you install them. There are limits to everything that has ever put to the tires you have to check that everything is clear.

until they are 16 "

You can, but check if you have ABS or not. Most ABS systems have a 5% margin of error built in and can get twisted with a tire size different than what they are calibrated.
Changing the diameter of the tire will also change your speedometer reading (wider tires read more slowly than does travel, because they take fewer rounds to cover the same distance).