Suppose that $X$ is a general $(4,1)$ hypersurface in $\mathbb P^3 \times \mathbb P^1$, which we think of via $\pi : X \to \mathbb P^1$, the projection onto the second factor, as a family of K3 surfaces.


For a genenal $t \in \mathbb P^1$, $X\_t$ doesn't contain any $(-2)$ curves, but for special values of $t$ it does (for example, if the quartic contains a line). Suppose that $C \subset X\_t$ is a $(-2)$-curve on $X\_t$. What is $N\_{C/X}$?


Of course it is an extension of $N\_{C/X\_t}$ by $N\_{X\_t/X}\vert\_C$, i.e. of $\mathcal O\_{\mathbb P^1}(-2)$ by $\mathcal O\_{\mathbb P^1}$. But that could be either $\mathcal O\_{\mathbb P^1}(-1)^{\oplus 2}$ or $\mathcal O\_{\mathbb P^1}(-2) \oplus \mathcal O\_{\mathbb P^1}$, and I don't know how to distinguish.



Normal bundle of "extra" curve in a fiber
If you want to check the condition in practice, you can consider the following exact sequence
$$
0 \to N\_{L/X} \to N\_{L/\mathbb{P}^3 \times \mathbb{P}^1} \to N\_{X/\mathbb{P}^3 \times \mathbb{P}^1} \vert\_L \to 0.
$$
The second term is $\mathcal{O}(1) \oplus \mathcal{O}(1) \oplus \mathcal{O}$, the third term is $\mathcal{O}(4)$, and if $F(x\_1,\dots,x\_4,y\_1,y\_2)$ is the equation of $X$, and the line $L$ is given by $x\_1 = x\_2 = 0$ and $y\_1 = 0$, the map is given by
$$
\mathcal{O}(1) \oplus \mathcal{O}(1) \oplus \mathcal{O} 
\stackrel{(\partial F\_/\partial x\_1,\partial F\_/\partial x\_2,\partial F\_/\partial y\_1)(0,0,x\_3,x\_4,0,1)}\to \mathcal{O}(4).
$$
If it is an isomorphism on global sections, the kernel is $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$, otherwise it is $\mathcal{O}(-2) \oplus \mathcal{O}$, or even $\mathcal{O}(-3) \oplus \mathcal{O}(1)$. 


