So if I understand the OP correctly this is the situation he/she describes:
Two groups were selected in some way so that the individuals in group A were all healthy, whereas the individuals in group B were all ill. In each of the two groups, an outcome variable was measured, call it $Y_0$. Then a treatment of some kind was given in both groups. After waiting a while post treatment, the outcome variable was mentioned once more, $Y_1$.
The researchers then calculated $\Delta Y = Y_1 - Y_0$ for each individual in each of the groups. Then, t-tests where done to see if $E(\Delta Y_A) = E(\Delta Y_B)$, where $\Delta Y_A$ and $\Delta Y_B$ is the difference in outcomes within group A and B respectively. Furthermore, t-tests were done within each group and on the pooled data. Thus the four hypotheses tested are:
- $E(\Delta Y_A) - E(\Delta Y_B) = 0$
- $E(\Delta Y_A) = 0$
- $E(\Delta Y_B) = 0$
- $E(\Delta Y) = 0$
I assume that the tests were chosen because of the belief that they correspond to the following following hypotheses, if rewritten in words:
- In the best case, you would there test the hypothesis of "do healthy people react differently to the treatment than ill people do?".
- Do the healthy people react to the treatment?
- Do the ill people react to the treatment?
- Do people, overall, react to the treatment?
However, there are some very important things to point out here. First and foremost, since there is no control group, hypotheses 2-4 can not be separated from questions such as "do people get better over time regardless of treatment or not?" since there is no data for what happens when an individual is not treated. This is also related to the regression to the mean problem that can can arise from the fact that the healthy and ill individuals must, in some way, have been determined as healthy/ill prior to this. See for instance Senn (2011) or Bland & Altman (1994) for an introduction to regression to the mean.
There might be some merit to the first hypothesis, but even there we can not separate the question "what is the difference in treatment effect between the two groups?" from the question "what is the difference in effect of time between the two groups?" since we once again do not have non-treated individuals. However, it is very likely that the healthy individuals differ from the ill individuals, meaning that unless you adjust for those differences (for instance in a regression model) you will get a bias due to this too.
It's worth pointing out that there are designs where valid inference can be drawn despite that everybody receives the treatment. The AB/BA cross-over design is such a design. There each individual is exposed to the treatment and non-treatment conditions, but the order in which the two conditions are faced is randomised. Due to this randomisation we can separate the effect of period from the effect of treatment. In the situation you describe, the sequence in which each individual is treated is the same, making such a separation not possible. To learn more about the AB/BA cross-over design, see for instance Senn (1994, 2002) or Jones & Kenward (2015).
Bland, J. M., and Altman, D. G. (1994). Statistics notes: some examples of regression towards the mean. Bmj, 309(6957): 780.
Jones, B. and Kenward, M. G. (2015). Design and analysis of cross-over trials. CRC Press LLC, Boca Raton, FL. Third edition.
Senn, S. (1994). The AB/BA crossover: past, present and future?, Statistical Methods in Medical Research, 3: 303-324.
Senn, S. (2002). Cross-over trials in clinical research. John Wiley & Sons, Chichester. Second edition.
Senn, S. (2011). Francis Galton and regression to the mean. Significance, 8(3): 124-126.