Do I need to keep more than one of each dragon type?

More and more, I find the answer is yes, for several reasons. I somewhat disagree with Jason that you don't need to keep two at all times - it is more nuanced than that.

1. Available breeding time is sometimes very short. Blue Moon was only available for 72 hours, for example. If you don't have the right dragons on hand for one of these short windows, you might be out of luck, unless you plan to spend gems.
2. Getting two of all Limiteds (other than Gemstone) is essential. Once they are gone, they're gone - but a breeding pair can produce a whole park full if there is an advantage to do so.
3. Getting two dragons let's you make eggs. By the time you hit the highest level, if you've stuck with the game that long you'll likely think about making some kind of egg display extravaganza. Trying to make an egg for any of the Epics will be quite hard without a breeding pair.
4. The rules of the game change frequently. Until recently, there were no Large Sun and Moon Habits - making a stockpile of Sun/Moon dragons relatively unnecessary unless you had no Rainbows. Now the Large Sun/Moon habitats are the best earners - but only if you can fill them each with three Sun/Moon dragons.

But EBongo, my hibernation cave is too small !

Well, there's the rub. It is one of the reasons to prioritize hibernation cave upgrades, but in the early going you are going to be pressed for space. When in doubt, make space for a pair in this order:

1. Limiteds (other than Gemstones)
2. Epics
3. Opposites/Uncommons
4. Commons
5. Gemstones (once your Gemstone habitats are full, there isn't much point to keeping them)

When you have two breeding locations and you are trying to obtain a time limited dragon that can only be bred via two specific dragons, having multiples of the required dragons allows you to make two breeding attempts at the same time.

However, you don't have to keep two of every dragon at all times - you can just breed the extra when needed and then stash them in the Hibernation Cave when you are done with them.

Also, as Peter mentioned in his comment, having two time limited dragons allows you to continue breeding those dragons (for fun or profit) after they are no longer available.

Well my answer would be a 'yes' but also a 'no' because if I breed 2 limited dragons or so I would keep them because there limited and after there gone...there gone! But if its just a poison dragon for example then no I wouldn't breed another one and if I did I'd just sell it because I've already got one...if you know what I mean. Anyway hope this helped a bit more and cya! :)