Personally, I fall into the latter camp; I can play one role well, two less proficiently, and the rest very poorly. There's a big reason that works OK for me, but almost everyone will want to be diverse. You simply won't be able to play a huge percentage of ranked and normal games in one role.
Mastering a single role
Before we go too far, I'll say that I'm not really a master of anything. However, I have more games as jungler than all other roles put together now (excluding Crystal Scar games where everyone is a roaming ambusher), and I am definitely better at it than I am in other roles. My map awareness is better, my timings are better, and my specific situational knowledge is much, much better.
That's really what experience gives you over someone else: specific situational knowledge. If you play one role a lot, you start to notice habits of specific types of players. You start cataloguing specific ways that specific characters get played. You start seeing patterns of ward placements, timing of aggression, and aggressive movements.
For instance, I am much better at predicting the enemy jungler than my average opponent. I rarely get counter-ganked at my skill level, even when I'm moving in a way that I think is obvious. On the other hand, I usually land at least one counter-gank per game, and I almost always mount at least one major invasion into the enemy jungle with the intent of killing him. I might not be better at executing these things than an average player, but I know and understand them when my opponent almost never tries to contest my ability to farm the jungle or kill me during camps. Relative to my skill level, my knowledge of these things is pretty high.
More important than that is knowledge of weird situations. Every few games, something odd happens. Top lane accidentally steals his team's first blue buff, or your team repels an invasion but your jungler is very low HP and has to start his route late when monsters are much stronger. Sometimes you're mid against Urgot. You just never know.
The more situations like these you get exposed to, the more you learn to handle them when they come up. Players that learn to handle these weird situations end up a lot better off for it. Unfortunately, you can't learn them by practicing, and you can't learn them by playing other roles. Weird situations that happen to the support are very different from weird situations that happen to top laners.
It's important to be able to diversify regardless of your mastery of other lanes, but playing every role is a viable option unless you're intending to be a pro-gamer. If you're a midlane expert, expect to be upset often when someone steals your role. It's impossible to play every game in mid, so you have to learn something else. If you can play any role, you'll come off as more cooperative and helpful, and that can really help your team win.
If you play multiple roles, however, you're really going to have to put more time into the game. While playing any role helps you develop basic mechanical skills and game sense, it still takes practice to learn specific lane gameplay. This is especially true of mid lane.
The upside is that having higher basic mechanical skills and game sense really does help. If you've played a lot of games and understand the game well in general, it helps you understand the big macro situations and make plays for your team. The elementary things like watching the minimap become a lot easier when you practice them over many games, and that awareness is often not present in self-proclaimed "mid experts."
Role priority and popularity
One thing that determines other roles you should play is popularity. If you main mid lane, you should learn several other positions. If you main jungle, you can almost ignore playing other positions.
Top and mid are the most requested. It feels as if top lane is the highest priority lane based on passive observation, but that's just my experience. Mid is the stereotypical first-picked lane. If you main one of these roles, you are going to be a jack-of-all-trades. In ranked games, you're guaranteed your preferred role only if you get a top slot.
Marksman/ADC and support are roughly even right now. Support might even be more popular than marksman now, as I mentioned last week. Marksman has suffered a dip in popularity after the Corki/Vayne/Jinx nerfs and the new item changes pushed bottom lane's dominant role to the support. I'm not really sure which one is higher demand; I haven't been keeping much of a finger on the pulse of this. It seem as if support is more requested now, whereas before ADC was nearly as high demand as mid or top.
Jungle has always been low demand. I think that people misunderstood just how often last pick players ended up jungle in ranked games. Sure, support had that rep for a long time, but jungle has always been the role that people don't understand. People who have never jungled find it intimidating to learn, and the whole role feels very alien compared to the relative stability of other roles. Because I'm a jungle main, it's easy for me to notice that my role is rarely contested.
The above facts make it very possible for me to almost always play something I like. I hate solo lanes, and I like jungle most. I've played exactly one game in recent history as anything other than jungle or support, and it was mid lane. Unsurprisingly, I got wrecked and we lost badly. In a weird twist of fate, the solo lanes were the last picked, and neither of us wanted them.
Fortunately those games don't happen often. However, as in all unusual circumstances, it pays to have a bit of preparation. While I have a pretty big library of unusual happenings that occur while playing the jungle, I don't really have an answer for the time when I am forced into a solo lane. That's definitely a weakness worth improving on.
We understand what it's like to climb the skill ladder in League of Legends. The Summoner's Guidebook teaches you the tools you need to get a competitive edge. Whether you're climbing the ranked ladder, playing Draft Dominion, or getting crushed by intermediate bots, every enemy has a weakness. And every Thursday, Patrick Mackey shows how you can improve improve on yours.