So, there was apparently a bit of a (possibly engineered) storm over on the boards yesterday about a 4E GM saying that players couldn't play Dragonborn in his game. This was waved around as an example of totalitarian GMing, and the snippets I picked up were enough to convince me to stay the hell away from the conversation, but it's an interesting question, all the same.
I have exactly no information about the original situation, but I can conceive of a few scenarios which I would interpret very differently. On one hand you have a GM who simply does not like the race, and is unwilling to talk about it. That's not great - 'I don't wanna, and I won't talk about it because I'm DM!' is pretty lame on the face of it.
On the other hand, if the DM had said "We're playing a game in an established setting (like Middle Earth). There aren't any dragonborn in the setting, so they're not a viable player choice" that's not such an unreasonable position, Especially if the GM is willing to discuss things.
I call this a Pendragon Ninja problem because that's probably the most succinct way to illustrate the disconnect, as in:
Bob: I'm running Pendragon. Who's interested?
Tom: Oh, yeah! I want to play a ninja!
I like this example because the problem is obvious (while the Dragonborn one is potentially muddled), which makes it easy to move on to the potential solutions.
First off, a lot of GMs will be comfortable just saying "No" to a request like this. I won't come out an endorse this because I'm one of those guys who likes to talk things out, but sometimes it really is the right answer, especially if you know your players well enough to know that Tom is taking a piss or will be equally enthusiastic about his next idea. The failure of advice will always be that you know your group best, so if you're really comfortable just saying no, then go forth and rock on. But if you're not, there are a few other approaches.
First, try to figure out what about the idea appeals to them. For Tom, is it that he wants to play an outsider? A sneaky character? An assassin? It's possible that he could play a character of that type that's still within the bounds of the premise. Playing a saracen might capture the outsider vibe, playing a knight with "off the books" skills might capture the sneaky/assasin part. Try to find the things that make the idea cool to Tom and see if you can pitch ideas that capture those elements.
Of course, that won't always work. Tom might just really be in it for the black pajamas and shuriken. In that case, ask yourself if the setting can deal with the idea as an exception - is it possible this is the only one in the world, so to speak. This can be easier to justify in magical games as the character is form another world or era, transported here through a freak accident, but you can usually find some way to make it work. Some GM's buck as this idea as violating the premise, but consider that a lot of fiction makes good use of these fish out of water ideas, and it's not unreasonable for players to seize upon them.
One catch is that it makes the fish out of water element front and center on the character. The lone ninja in the arthurian court is count to stand out as FOREIGN and DIFFERENT and a lot of interactions will center on that difference, and that may not be what the player is interested in. Perhaps even more importantly, such a character can easily become the lynchpin of a game unless you are careful in your handling of it. Maybe that's a great idea for your group, but if Tom is not the guy you want to hang the game off of, then you need to be careful to keep it from being the adventures of a Ninja in King Arthur's Court.
So, that's nuance, and that doesn't always fly. Tom may not want to have to deal with those issues - he just wants to have cool fights with ninja weapons, and Bob is having trouble explaining to him that these aren't anime fights, and ninja vs. knight may not go exactly the way he imagines. So Bob is pretty much down to three options.
First, he can just say "no". He's made a good faith effort to accommodate Tom, but it's just not working. No harm, no foul, just try something else.
Second, he might suggest that Tom would not enjoy the game. He really should try saying "no" before jumping to this, but after a few no's, this might be the only options.
Third, he could consider changing the game. Depending upon what excites Bob about running pendragon, he might be able to switch to a game like Legend of the Five Rings or Blossoms Are Falling and keep everyone happy. This is not a trivial consideration though: the amount of work Bob will need to do to shift gears is an order of magnitude greater than what it would take for Tom to tweak his character. More importantly, if this idea doesn't excite Bob, then it's a bad idea. If the GM isn't excited about the game, then it's very near DOA.
There are a lot of issues that run through this topic, ones of player respect, GM empowerment, theme protection and god knows what else. The good news is that the theory is more convoluted than the reality. As a hypothetical problem, this can be daunting, but as an actual problem you're talking to your friend about? You'll be amazed how easily it clears up.
1 - 4e introduces another layer to this since the objection (and the desire t play a Dragonborn) may have been entirely mechanical. Thankfully, 4e also provides excellent tools for dealign with this, since it's trivial to re-skin the race. Thus, for example, if the DM objected to the Dragonborn's breath weapon as unbalanced, he could include Dragonborn, but use the Goliath or Hobgoblin racial modifiers to represent them mechanically. Alternately, if the player explicitly wants the mechanic, it would be possible to make the character nominally a member of another race but use the Dragonborn abilities, possibly reskinned as a war shout or magical ability.
2 - Pendragon, if you're not familiar with it, is a game that is explicitly about playing ladies & knights in the Arthurian period. There's some wiggle room around that, but it's really all about ladies & knights.
3 - Yes, I know this is terrible. No is such a negative, unfriendly, non-communal word. A lot of us are geeks, and we want everyone to get along, and we don't like drawing hard lines as a result. I sympathize, I really and truly do, but at some point you need to put on your big girl panties and do it. You'll be amazed to discover that it does not actually end in tears and door slamming most of the time.
4 - This is actually a somewhat contentious point. I'm assuming in this case that we're talking about very traditional games where the GM puts in a lot of work to make the game happen. Setting aside the issue of authority within the game, that raises the question of how much the GM can reasonably demand. Some people feel that the extra work entails a bit more weight to his opinions, while others seek something more egalitarian. This gets further muddled when you have groups with multiple GMs, since the decision is no longer between "Game or not game" but between "My game vs. his game" and at that point it ends up looking like.... well, I suspect you can imagine it.