Since Flappy Bird was taken down from the app store, tens of similar games have popped up. Two of which are currently the top free games on the app store!
Is it legitimate to piggy back off Flappy Bird's name like this? The new games that are copying Flappy have names like "Splashy Bird" and moreover the graphics and concept of the games are basically all the same.
Enforcing trademark issues is very hard, especially when it comes to international markets and laws. In theory, the name shouldn't be too similar while product is essentially the same, but it's all arguable. I think a big consideration is how big of a company are you going up against - you wouldn't dare to copy products of big companies. My impression is that big companies get away with more in these cases because of legal muscle in the battle.
In the case of new games that are rip-offs of Flappy Bird, the biggest risk is that if creator of Flappy Bird sells his trademark and rights to a company who would be willing to go after all the imitators for share of their profits.
Like @webbie said, it can come to bite you if the copyright owner has deep pockets.
One way around if your app qualifies as a parody. Because parodies are "fair use":