Strange question. If you were dreaming that you botched on a psionic skill that you really had, could you feel the effects in the real world? It IS a psionic skill controlled by your mind, after all. Doesn't really require a lengthy explanation of Da Rulez!!!!! or anything.
(Doing Therapy)
I would imagine this is very rare, because there is a difference between thinking you're doing something and doing it. You can dream that you're doing math, and math can be a thing that you can do with your mind while awake, but that doesn't mean the math you're doing in your dream makes any sense at all.
I'd say most psionic skills require a certain level of awareness to perform that being asleep would prevent.
I'm sure this is possible, though. It'd be about as rare as someone sleepwalking to their private jet and sleep-flying to Cuba, though.
Dreams are a strange thing though. I had a dream once that I was falling, and I woke up when I "hit the ground" and I could have passed a lie detector that I had actually fallen from some height back into my bed.
(Doing Therapy)
I'd go with Scott on this one. I can imagine someone designing a psionic skill that was supposed to work while they slept (e.g., some form of mental protection or an out-of-body or non-corporeality skill), but I would still expect the skill to be initiated while the character was awake.
I know that Eric once did battle on something like the dream plane, and I can perceive a reality in which the dream plane is a related universe and some characters are able to take action on it, but those characters have a particular skill that enables them to retain consciousness while sleeping.
--M. J. Young
I'm planning to design a skill which will let me train myself whilst I'm dreaming. It would be an example bonus, I think.
(Doing Therapy)