The Pokedex did stop talking about being weak in Silver, Soul Silver, Moon, and Ultra Moon. But in the other games, this might be why:
So far, however, we haven't really tied the Asian carp to the Magikarp we're familiar with. They're anything but dangerous, being the original "joke" Pokémon. Their only move until Level 15 is the useless Splash. After that, they can learn Tackle and Flail by level-up, and sometimes Bounce via other means. But there's a pattern here, and it becomes evident when you realize that the Japanese name for Splash, haneru (はねる) might be better translated as "hop": this is why Pokémon like Hoppip, Spoink and Buneary, who have no connection to water, can learn it too. Most of Magikarp's moves – hopping, flailing and bouncing – allude to the carp's jumping ability. The jumping is the key, because the sight of jumping carp inspired a Chinese legend that, in turn, inspired Magikarp.
The story went that if a lowly carp could leap over a particular waterfall, referred to as the Dragon Gate, it would be transformed into a dragon, and that when one succeeded, the summer rains would start to fall. There are numerous waterfalls in China which are claimed to be the Dragon Gate, but the most notable is on the Yellow River, near Henan.
Carp are known for instinctively swimming upstream, and a waterfall would represent a clear barrier to further progress, so it's easy to see how the sight of jumping carp flinging themselves at the base of a waterfall might have inspired this legend. But what's really of interest is the fact that this legend is commonly used as an analogy for problems faced by us humans.
Traditionally the carp leaping the waterfall and becoming a dragon was associated with social climbing, especially in relation to passing the imperial examination, a challenging series of tests for entry into Imperial China's civil service, and one of the main ways in which ordinary people could rise to positions of real influence. Even today, it's said that students taking their exams are like carp trying to leap the Dragon Gate: it's a challenge, but the rewards will be great.
Similarly, while Magikarp is an overwhelmingly common and ordinary Pokémon, with enough effort it too can exceed expectations and become a dragon (though not a Dragon, to the continual annoyance of some fans). Incidentally, I can't be the only person to have assumed Magikarp's name – along with its Japanese name, Koiking – was a bit of deliberate irony, especially given that this Pokémon is often sold to the player by a smooth-talking salesman extolling its virtues. Having looked more into the legend, though, I'm not so sure. In illustrations of the Dragon Gate myth, the fish depicted are often majestic-looking koi carp, most frequently red or gold in color. This matches Magikarp's appearance in some respects (red and gold with a crown-like dorsal fin) though perhaps not all: Magikarp's vacant expression and habit of flopping around uselessly don't exactly embody any kind of regal dignity. So perhaps both interpretations are true. Magikarp, just like carp of legend, has the potential to become something magnificent through enough effort and perseverance. But in the meantime, it’s just a regular old fish, and you really can't expect too much from a regular old fish.