To determine the positive integers $n$ such that the expression

$$\dfrac{n^3 + 3}{n^2 + 7}$$

is a positive integer, we need to analyze when the expression simplifies to a whole number.

### Step 1: Dividing the Polynomials

Consider the division:

$$\dfrac{n^3 + 3}{n^2 + 7} = q(n) + \dfrac{r(n)}{n^2 + 7},$$

where $q(n)$ is the quotient and $r(n)$ is the remainder when $n^3 + 3$ is divided by $n^2 + 7$.

Perform polynomial long division of $n^3 + 3$ by $n^2 + 7$:

1. Divide the leading term $n^3$ by $n^2$ to get $n$.
2. Multiply the divisor $n^2 + 7$ by this term $n$ to get $n^3 + 7n$.
3. Subtract to find the new dividend: $n^3 + 3 - (n^3 + 7n) = -7n + 3$.
4. Divide the leading term $-7n$ by the leading term $n^2$, which is 0, so we stop here.

Thus, the quotient $q(n) = n$ and the remainder is

$$r(n) = -7n + 3.$$

So,

$$\dfrac{n^3 + 3}{n^2 + 7} = n + \dfrac{-7n + 3}{n^2 + 7}.$$

For the expression to be an integer, the remainder must be zero:

$$-7n + 3 = 0.$$

### Step 2: Solve for $n$

Solving the equation:

$$-7n + 3 = 0 \implies n = \frac{3}{7},$$

which is not an integer. However, since we need the entire expression to simplify to a whole number, check the divisibility condition for other values by ensuring $-7n + 3 \equiv 0 \pmod{n^2 + 7}$.

### Step 3: Check Small Positive Integers

We'll verify for small values of $n$ manually:

- **$n = 2$:**

$$\dfrac{2^3 + 3}{2^2 + 7} = \dfrac{8 + 3}{4 + 7} = \dfrac{11}{11} = 1,$$

which is an integer.

- **$n = 5$:**

$$\dfrac{5^3 + 3}{5^2 + 7} = \dfrac{125 + 3}{25 + 7} = \dfrac{128}{32} = 4,$$

which is an integer.

### Conclusion

Upon verifying integer values for $n$, we determine that the possible positive integers satisfying the condition are:

$$n = 2 \text{ and } n = 5.$$

Hence, the positive integers $n$ for which $\dfrac{n^3 + 3}{n^2 + 7}$ is an integer are:

$$\boxed{2 \text{ and } 5}.$$