* Your tokens can be tainted, for instance because they were once owned by a blacklisted address. The value of your tokens will be effectively 0.
* The issuer might run a fractional reserve (this is the accusation against Tether, the most popular stablecoin at the moment)
* The asset backing the stablecoin might not be backed by anything. It's not worth getting into FED policy here, but let's just say that a stablecoin denonminated in Venezuelan Boliviar is not attractive.
> Stablecoins seem like they would achieve most of the goals of the original cryptocurrencies such as Bitcoin
The original goals of cryptocurrencies (Bitcoin, to be precise) were exactly to avoid the problems that stablecoins have.