In the context of hypothesis testing using a paired t-test, if n is the number of pairs in the two related samples, the degree of freedom is

For a paired t-test with n pairs, the degrees of freedom equal n minus one.

A paired test works on the n difference scores, one per matched pair.

One degree is lost in estimating the mean of those differences, leaving n minus one.

The values two n minus one and two n minus two apply to independent-samples designs, not paired data.

Using n itself ignores the constraint imposed by estimating the mean.

The paired t-test suits before-after or matched observations.