I'm a vim newbie setting up some plugins. Some recommend installation via Vundle, others via pathogen. Is it possible to use both package managers at the same time in this circumstance? If so, is it just a matter of installing both and installing the relevant plugins according to their normal instructions, and nothing will conflict? Or do I have to do something special to make this work?
TL;DR: You don't need both; Vundle includes the Pathogen functionality (and more)
The Pathogen plugin just automatically extends your 'runtimepath' (locations where Vim searches for plugins) to include all subdirectories in ~/.vim/bundle/. This is simple, but allows to separate each plugin into a dedicated tree, making updates (e.g. via Git) and removals simpler.
Other plugins have built upon that idea and extended it with automatic download, installation, and updating, usually via some Plug foo/bar commands in your ~/.vimrc. As they need the same separation as provided by Pathogen, such implementation usually is included in them. (The core is just a one-liner, not much of a deal.)
You don't actually need any package manager; I personally use 100+ plugins without one. It's a matter of how often you use new machines with Vim (the central configuration and installation capabilities certainly make bootstrapping easier), and how often you update / try out new plugins. If you just use one or two systems, and are conservative in the number and update frequency of plugins, putting everything into ~/.vim/ might also work for you. (This is just to avoid that you're intimidated by those package managers; try them out, and use one if you like it.)