To evaluate the expression
\[ 6+\dfrac{4}a+\dfrac b3 \]
when \(a=4\) and \(b=3\), we start by substituting these values into the expression.
Substituting \(a=4\) into \(\dfrac{4}{a}\):
\[ \dfrac{4}{4} = 1 \]
Next, substituting \(b=3\) into \(\dfrac{b}{3}\):
\[ \dfrac{3}{3} = 1 \]
Now, we can substitute these results back into the original expression:
\[ 6 + 1 + 1 \]
Now we can perform the addition:
\[ 6 + 1 = 7 \]
\[ 7 + 1 = 8 \]
Thus, the evaluated result of the expression is
\[ \boxed{8} \]