$\because x+y=4$，
$\therefore x=4-y$，
当$y=1$时，$x=4-1=3$，
当$y=2$时，$x=4-2=2$，
当$y=3$时，$x=4-3=1$，
所以方程$x+y=4$是正整数解是$\left\{\begin{array}{l}{x=3}\\{y=1}\end{array}\right.$，$\left\{\begin{array}{l}{x=2}\\{y=2}\end{array}\right.$，$\left\{\begin{array}{l}{x=1}\\{y=3}\end{array}\right.$，共$3$个，
故答案为：$3$.

$\because x+y=4$，
$\therefore x=4-y$，
当$y=1$时，$x=4-1=3$，
当$y=2$时，$x=4-2=2$，
当$y=3$时，$x=4-3=1$，
所以方程$x+y=4$是正整数解是$\left\{\begin{array}{l}{x=3}\\{y=1}\end{array}\right.$，$\left\{\begin{array}{l}{x=2}\\{y=2}\end{array}\right.$，$\left\{\begin{array}{l}{x=1}\\{y=3}\end{array}\right.$，共$3$个，
故答案为：$3$.