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 conditions on the normal bundles are automatic (since they are a quotient of the tangent bundle of $\mathbb{P}^n$, which is ample). Hartshorne's conjecture has been proved by Lubke for any homgenous variety, so in particular for any grassmannian. See
Martin Lübke, Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten., Crelles Journal, 1980.
The general Hartshorne's conjecture is however still open, although there is some evidence supporting it. See the recent paper by Peternell: Submanifolds with ample normal bundles and a conjecture of Hartshorne