As Franscesco's answer shows, there are counterexamples to your problem.

However, if you put some mild restrictions on the subvarieties, the answer is yes. 

One of Hartshorne's old conjectures states that 'if $X$ and $Y$ are subvarieties of $Z$ with ample normal bundles $N_{X|Z}$, $N_{Y|Z}$ and $\dim X+\dim Y=\dim Z$, then $X$ and $Y$ have non-empty intersection. In $\mathbb{P}^n$ the condition on the normal bundles are automatic (since they are a quotient of the tangent bundle of $\mathbb{P}^n$, which is ample), and explains why the result is true in the $\mathbb{P}^n$ case.

Hartshorne's conjecture has been proved by Lubke for any homgenous variety, in particular any grassmannian. See

[Martin Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten.][1], Crelles Journal, 1980.

For some evidence for the Hartshorne conjecture, see the recent paper by Peternell: [Submanifolds with ample normal bundles and a conjecture of Hartshorne][2]


  [1]: http://www.reference-global.com/doi/abs/10.1515/crll.1980.316.215
  [2]: http://arxiv.org/abs/0804.1023