  A fullness conjecture of Kuznetsov says that if a smooth projective variety
$X$ admits a full exceptional collection of line bundles of length $l$, then
any exceptional collection of line bundles of length $l$ is full. In this
paper, we show that this conjecture holds for $X$ as the blow-up of
$\mathbb{P}^{3}$ at a point, a line, or a twisted cubic curve, i.e. any
exceptional collection of line bundles of length 6 on $X$ is full. Moreover, we
obtain an explicit classification of full exceptional collections of line
bundles on such $X$.
