The answer is no, since the infinitesimally deformations can be obstructed.
Mumford gives an example of a smooth surface $X$ containing a curve $C$ such that $h^0(N_{C/X}) \neq 0$ (and so $C$ can be deformed in $X$ infinitesimally at the first order) but $C$ cannot be moved globally; in other words, no effective cicle in $X$ arises from the first-order deformations of $C$.
The reduced subscheme of the Hilbert scheme $\textrm{Hilb}_X^C$ in a neighborhood of the point $\xi$ corresponding to $C$ consists only of $\xi$, but the tangent space of $\textrm{Hilb}_X^C$ at $\xi$ is $1$-dimensional. This means that the local ring of $\textrm{Hilb}_X^C$ at $\xi$ contain nilpotents elements, and shows that the appearance of nilpotents is unavoidable also in entirely "classical" questions of algebraic geometry.
For further details, see Shafarevich's book Classical Algebraic Geometry 2: Schemes and Complex manifolds, page 111.