fall down vs. fall through

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Fall down is wrong, there is another problem:

Because of heavy rain, our plans for a picnic fell through.

Thank you.

Just one more question: is it generally incorrect to write "plans fell down" or is it wrong only this case?

I can't think of a case where it would be correct to say "plans fell down."

For example:
Where do you think the plan falls down?
Source: http://dictionary.cambridge.org/define.asp?key=27746&dict=CALD

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Ah, I see. Yes, that is a good use, but there, 'fall down' means 'to perform disappointingly'. The plan has a defect within itself. In your original example, the rain (an outside force) ruined the plan, so 'fall down' does not work.
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plan ... falls through = the planned event never happens; the event is cancelled

plan ... falls down = the plan is ineffective; it doesn't seem like it will work if implemented
CJ