  An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an
interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one
edge of $G$ colored by $i$, and the colors of edges incident to any vertex of
$G$ are distinct and form an interval of integers. In this paper we prove that
if a connected graph $G$ with $n$ vertices admits an interval $t$-coloring,
then $t\leq 2n-3$. We also show that if $G$ is a connected $r$-regular graph
with $n$ vertices has an interval $t$-coloring and $n\geq 2r+2$, then this
upper bound can be improved to $2n-5$.

Discrete Mathematics