Degrees of freedom are not decoration inside the parentheses: they say how many cases entered that particular analysis. When they do not match the N in your Participants section, cases were lost and never accounted for.
Independent-samples t. Degrees of freedom are N − 2, because two means are estimated. t(98) means 100 cases entered that comparison.
Paired-samples t. They are N − 1, and N is the number of pairs, not observations. Confusing the two is one of the most common errors in pre-post designs.
One-way ANOVA. Error degrees of freedom are N − k, with k the number of groups. Effect degrees of freedom are k − 1, so F(2, 87) means three groups and 90 cases.
Multiple regression. Residual degrees of freedom are N − k − 1, with k the number of predictors. This is the one most often miscounted, because the −1 for the intercept gets forgotten.
Almost always, undeclared missing cases. If Participants says 120 and the regression has 104 residual degrees of freedom with three predictors, 108 people were analyzed: twelve dropped out and nobody says so. That is an omission rather than fraud, but it is exactly what a reviewer asks about.
Sometimes, listwise deletion. With several analyses and different missingness per variable, each model ends up with a different sample. That is legitimate, but it has to be stated with the N for each. Worked through in how to report missing data.
And occasionally, the wrong N. In repeated measures, reporting the number of observations instead of participants inflates the sample without the text showing it.
Degrees of freedom are the one figure in a paper that cannot be dressed up: they come out of the model that was actually fitted. That makes them the first place to look when something does not add up.
They also have a useful property: they are consistent across analyses. If three models on the same variables imply three different Ns with no explanation, there is something to be told there.
And if the p value does not match those degrees of freedom either, you have two signals on one line. That one is checked in the p value against its degrees of freedom.
In classical tests, yes: one degree is lost per estimated parameter. In mixed models, and under corrections such as Greenhouse-Geisser, they stop being whole numbers and stop being deducible that directly, and this check no longer applies.
They come from a correction: Welch when variances are unequal, or Greenhouse-Geisser and Huynh-Feldt when sphericity fails. They are correct and should be reported with their decimals, but they no longer let you infer N.
If it varies between analyses, yes. One sentence in Results or a table note saying how many cases entered each model and why cases were lost is enough. Staying silent is what turns a reasonable decision into a suspicion.
Yes, and it has to be declared with the criterion you used, decided BEFORE looking at the results. Covered in how to report outliers.