When editing Lisp code, can emacs be configured to display each nested level of parentheses in a different color?

I think you are looking for something like mwe-color-box.el

There is the package rainbow-delimiters, which doesn't do exactly what you want, but does just colorize the parentheses - which is a nice subset and allows you to still see the other syntax highlighting provided by the major mode.

If you need this to help editing, then I suggest turning on coloring the innermost sexp which contains the cursor with a different background color. At least I'm used to this and it is sufficient.