I currently have Unity and Gnome 3.6 installed on my computer and I want to check out the KDE desktop environment. Is there any problem with installing multiple desktop environments on my desktop?
You won't have any problems installing multiple desktop environments. Nothing in the system will stop you from installing KDE, GNOME, Unity, Enlightenment, and all the others all at once (except for disk space).
However, desktop environments will often "argue" with each other and overwrite settings. For example, installing KDE on a system will very often break a Unity installation by overwriting GTK or similar properties. Similarly, installing Unity will break KDE most of the time.
Unfortunately for you, these incompatibilities are very hard to repair after they've happened, so this is a risk you must be willing to take with your system. If you want to explore different environments, I'd highly recommend you use VMs until such time as you find one you like. Then, install that one and only that one on your system.
That is rather difficult to answer depending on what will be considered a problem and on your level of tolerance to what are arguably small or negligible problems. (Some people are very tolerant to errors in Linux just because they haven't payed for it.)
You can install as many desktop environments you want, but... is that reasonable? - Isn't that even crazy? - If the camel's back didn't brake it doesn't mean we have to try to break it.
There are also problems and bugs that can happen, and conflicts are also possible even without multiple desktops.
I think these risks will increase the more DEs are installed on the same system, the greater the difference is between them, and the heavier/complex they are.
The greatest difference I can see is that between KDE/Qt and GTK systems. Unity, Gnome and KDE/Plasma are all rather large and complex (compared to Xfce. LXDE, Openbox, Fluxbox).
As for my own experience, I have always had problems with multiple desktops, usually only in one of them. That is: there may be a big chance to have some problems at least in one of them.
I wouldn't dream installing as many desktops environments as possible, whatever that means.
Want KDE? Go for Kubuntu.
Want Xfce? Go for Xubuntu.
Want LXDE? Go for Lubuntu.
Want GNOME? Go for Ubuntu GNOME.
Want GNOME alongside Unity alternatively?
That seems reasonable, but some say no on that also: https://askubuntu.com/a/491049/47206.
Just to test them on the short term, one could go with a virtual environmen (VirtualBox), a bootable live-usb system, or even a separate partition in multi-boot. The latter is more complicated (you have to know what you do) but I prefer it to mixing DEs.
You won't have problems installing them, you might even be able to permanently change to a new DE, if you're ready to spend a relatively long time tinkering around and adjusting settings.
But desktop environments are big complex things designed to be usable, not swappable. So if all you want to do is check out another DE, you will get a better idea by running a distro that uses the DE of your choice (either in a VM or booting a live CD/USB)
You can install as many desktops you want, but only to test them.
Forget what everyone says.
An open source OS can be used the way you like it, that's why its open source. But to make something work correctly, you have to install it's ppa and all their dependencies too.
If you do that, then all desktop environments will work just fine. If you miss something , then yes you will have many issues.
Also about ram. All desktop environments won't be running at the same time. At login you choose one.
Disk space is the only thing you need. That's all.
There can be problems but depends on the desktop environment you are working on vs which you prefer to install. The main problem comes with the dependencies and default packages as many packages are same for different desktop environments.
I once installed pantheon desktop environment on top of an existing Ubuntu Unity installation, after installing the new pantheon desktop(which was in beta..loki early releases) was full of bugs and glitches and was not pleasant at all to use. So i decided to uninstall my pantheon and return back to unity. As it happens after purging pantheon daily ppa and its packages it broke my unity installation, almost to the point where i had to reinstall Ubuntu itself.
On a similar case i installed budgie-remix stable from ppa and wanted to try out its cool new looks, as it turns out again the experience was not quite upto my expectations so i purged and removed budgie-remix and it did not cause any problems with my desktop at all.
So, As long as the installation environment does not have any problems with the existing environment it's all good!.
P.s: These are my observations, you can do whatever the hell you want if you have some terminal experience up your sleeve.