The answer is no, in fact there exist examples of non-reduced projective curves which are non smoothable.
Perhaps the easiest example is the double line, i.e. the scheme $X=2L$, where $L \cong \mathbb{P}^1$. In fact, one can show that
- $H^0(X, T^1_X)=0$, so every deformation of $X$ over the dual numbers is locally trivial;
- $H^1(X, T_X)=0$, so every locally trivial deformation of $X$ is actually trivial.
It follows that $X$ is a rigid scheme. Since $X$ is projective, rigidity implies that for any flat family $Y \to T$ whose central fibre is isomorphic to $X$, nearby fibres are isomorphic to $X$ too (see [Hartshorne, Deformation Theory, Section 5]).
So $X$ provides a counterexamples to your question.
I do not know whether it is possible to give a counterexample with $X$ affine.