Jenny is eight years older than twice her cousin Sue’s age. The sum of their ages is less than 32.

Let Sue's age be \( s \). Then Jenny's age can be expressed as:

\[
j = 2s + 8
\]

According to the problem, the sum of their ages is less than 32, which gives us the inequality:

\[
s + j < 32
\]

Substituting \( j \) into the inequality, we get:

\[
s + (2s + 8) < 32
\]

Simplifying this, we have:

\[
3s + 8 < 32
\]

Subtracting 8 from both sides:

\[
3s < 24
\]

Dividing both sides by 3:

\[
s < 8
\]

Since \( s \) must be a whole number, the greatest possible age for Sue that satisfies this inequality is \( 7 \).

Thus, the greatest age that Sue could be is:

\[
\boxed{7}
\]