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I am wondering under which conditions the intersection of two smooth manifolds, say $X$ and $Y$, is a smooth manifold. I know that if they intersect transversally then the intersection is a manifold.

Now suppose that the manifolds do not intersect transversally. Can the intersection still be a manifold? And can the formula $\operatorname{codim}(X\cap Y)=\operatorname{codim}(X)+\operatorname{codim}(Y)$ still be valid?

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Yes, it sure can. Consider $X=\{y=0\}$ and $Y=\{y=x^2\}\subset\Bbb R^2$.

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    Thank you very much! For some reason I was not accepting that a single point is a manifold. Thank you! – 2017-01-21
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    Well, we can move this same sort of example into any number of dimensions you want :) – 2017-01-21
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    Just take $X = Y$, the intersection is the smooth manifold $X$. @dlc – 2017-01-21
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    @JohnMa: But that will violate the codimension condition the OP wanted. :) – 2017-01-21