No, bash can't emulate older versions of bash. But it's pretty easy to set up a test environment that includes an older version of bash.
Installing older version of individual software is tedious if you have to install each software package manually, not to mention resolving the library incompatibilities. But there's an easier solution: install an older distribution. Installing an older distribution, with a consistent set of software including development packages, costs about $1 of hard disk space and maybe an hour to set up the first time.
The schroot package makes it easy to install an older (or newer!) Linux distribution that's running on the same system as your normal Linux system. You can easily make a schroot setup where you can run a program in an environment (a chroot) where the system directories point to the older software, but the home directories are those of the normal environment. I wrote a guide for Debian-based distributionsa guide for Debian-based distributions; you can easily go back to Debian slink (with bash 2.01.01) this way.
If you want to test with different Unix variants, different CPU architectures, or very very old software, you can run other OSes in a virtual machine. There's a little more overhead (in RAM, disk space, CPU and maintenance) but it's still very much doable.