I recommend using any derivative of Debian, and using checkinstall or debuild to install any programs that aren't in the repos. Using these, you can keep your entire system under the control of the package manager, which should help prevent breakage. Here are some example use cases for each tool (packages and version numbers made up):
You're running Ubuntu 9.04 which has libsomething version 3, and you want to install something that requires libsomething version 4. First, check if the development branch (Ubuntu 9.10) has libsomething version 4 by searching http://packages.ubuntu.com/. If it does, you can try to download the deb file and install it.
This may not work, because the libsomething-4.deb from the devel repos might have been compiled with other updated versions of packages. If it doesn't work, you can download the source and use debuild to generate a deb. You may have to run apt-get build-dep libsomething first, in order to install the packages needed to compile libsomething.
To run debuild, simply unpack the source tarball, and cd into the resulting directory. Then run debuild, instead of your usual ./configure; make; sudo make install. If there are no errors, you'll be left with a deb file, which you can install.
If libsomething version 4 is not in the devel repos at packages.ubuntu.com, next try packages.debian.org. If that doesn't work, then you;ll have to download the source from libsomething's own webpage and compile it yourself, using checkinstall to install it. To do this, download the source tarball from the libsomething website and unpack it. Do the usual ./configure; make, but instead of sudo make install, use sudo checkinstall. This will install the program while keeping track of exactly what files it installs. Then it will make a deb file out of them and install that. This allows the source-installed package to be managed by the package manager.
In summary, apt provides many ways to accomodate source-built packages without breaking the package manager by installing things manually outside of its knowledge.