Psychics
Inability to Shut Off
Primarily a weakness for telepaths, pretty much all of those with psychic abilities are unable to turn their powers off. When first coming into their abilities, hearing so many thoughts that are not their own or watching items fly around whenever they get overly emotional can tend to take a mental toll.
Many a telepath winds up diagnosed with schizophrenia, and many a telekinetic can wind up thinking they are being haunted by a poltergeist.
With time, they may be able to dull down the voices or the physical activities with practice, but it's not something that happens overnight, and they can never really shut things off completely.
Telepathic Language Barrier
Just because a telepath is able to hear someone's thoughts, doesn't necessarily mean that they'll be able to understand them. If they hear someone thinking in another language (Japanese, German, etc.), and they themselves do not know the language, they will not be able to detect what it is the person is thinking exactly.1
Overwhelmed
If emotions are running high in a given situation, a sensitive psychic may be completely overwhelmed by what they are hearing and seeing. For a telepath, this can send their brain into an 'overload' situation, wherein they'll develop a nosebleed and/or begin to shutdown mentally into a semi-catatonic state. For a clairvoyant, they can become lost in the vision they are having and need an anchor, or another person to pull them out of it. For a telekinetic, they may be unable to control their abilities, period. Any extreme change in their emotional state can cause them to move things without realizing they're doing it.
