They do, actually; a brand new Core i7 still runs 16-bit real-mode and 32-bit real-mode.

Now, if you're running in 64-bit mode, there's no vm86 mode you can use to run 16-bit applications on a 64-bit kernel. However, the CPU still supports 16-bit real mode, and under a 32-bit kernel you can run 16-bit applications.

Edit: An edit went on the wrong comment somehow. Ignore what it said.

Here's a screenshot of Windows 8 (32-bit) running four different Windows 1.01 apps:

http://toastytech.com/guis/win816bit.png

Most people here seem to be talking about binary (.exe) compatibility.

I'm sorry if I was unclear in my wording, I mean the modern Windows version must be 32-bit to be able to run 16-bit Windows apps.

That's just not true. Windows 3.1 removed real-mode support in Windows itself -- in other words, the last version you can run on a 8086 CPU is Windows 3.0.

But pre-3.0 16-bit Windows apps certainly didn't stop working.

Also, present-day x86 CPUs definitely can run real-mode software using virtual 8086 mode: http://en.wikipedia.org/wiki/Virtual_8086_mode

This is the same mode that Windows 3.x used to run multiple real-mode DOS apps simultaneously on a 386 processor.

There were some exceptions to this; I remember as a kid upgrading my 486 from Windows 3.1 to 95 for the first time, and being disappointed that Setup replaced Paintbrush and Write with Paint and Wordpad, because I liked Paintbrush better. It actually replaced PBRUSH.EXE and WRITE.EXE with dummy EXE:s that would just launch MSPAINT.EXE and WORDPAD.EXE instead. These dummy files are still in recent Windows versions for compatibility with stuff like ancient apps that create shortcuts to launch Write with their docs.

It was trivial to retrieve PBRUSH.EXE from Windows 3.1 install media, however, and it works fine on Windows 95 as well as every subsequent 32-bit Windows.