Benjamini-Hochberg procedure (FDR) generally assumes tests are independent from each other (but see this), so it may not be an ideal test for among-conditions assessment (the 6 pairwise comparisons). Otherwise, 'to run a bunch of tests' sounds like a single-family/question approach so you'd need to apply correction to the entire population of p-values generated. In other words, you just set your bet 180 times, chances you 'win' in this roulette are quite a bit higher than if you were to set 1 bet. A more cohesive design e.g. a two-way ANOVA with Tukey-Kramer post-hoc comparisons, assuming the 4 conditions are the same throughout, would be much preferred.
It's not only possible, it's explicitly recommended (see Ruxton & Beauchamp 2008).
First, if the omnibus-test of homogeneity across all groups is not of interest (e.g. the overall ANOVA $F$-test), consider not paying attention to it at all or not doing it in the first place. Second, if each planned comparison tests a different specific hypothesis, it is actually controversial if a formal control of the experimentwise type 1 error rate (EER) is required or not. Some text do not consider it necessary (Kirk 1995, Quinn & Keough 2002, Rothman 1990, Rubin 2021, Sokal & Rohlf 1995). Ruxton & Beauchamp (2008) recommend not controlling EER if the set of pre-planned contrasts is orthogonal. If all possible pairwise comparisons between groups are planned, the set of contrasts is not orthogonal and so, Ruxton & Beauchamp (2008) recommend controlling EER.
Rubin (2021) on the other hand argues that in the case of individual testing, no alpha adjustment should be done. He defines individual testing as tests, where each individual result must be significant in order to reject each associated individual null hypothesis. This seems to be the case here if you want to test each pairwise group difference individually. Personally, I find his arguments convincing and have subsequently changed my own opinion on the matter.
References
Kirk RE. 1995. Experimental design. Pacific Grove (CA): Brooks/Cole.
Quinn GP, Keough MJ. 2002. Experimental design and data analysis for biologists. Cambridge (UK): Cambridge University Press.
Rothman, K. J. (1990). No adjustments are needed for multiple comparisons. Epidemiology, 43-46. (link)
Rubin, M. (2021). When to adjust alpha during multiple testing: A consideration of disjunction, conjunction, and individual testing. Synthese, 1-32. (link)
Ruxton, G. D., & Beauchamp, G. (2008). Time for some a priori thinking about post hoc testing. Behavioral ecology, 19(3), 690-693. (link)
Sokal RR, Rohlf FJ. 1995. Biometry. 3rd ed. New York: WH Freeman.
Best Answer
As Glen says, it depends on your hypotheses, and especially the relative importance of Type I vs Type II errors. For example, if you're testing a new drug, you don't want to risk missing side-effects because they were rendered nonsignificant by the multiple comparison correction. A similar principle applies when testing for effects of contaminants in the environment. If you're data mining without strong a priori expectations, you might be more inclined to control the Type I error rate to ensure that any 'interesting' results you find are genuine.
Bonferroni in particular has fallen out of favour because it is very conservative (i.e. by controlling false positives you're drastically increasing the number of false negatives). Other options are the Sidak correction (which is less conservative when the family of comparisons is large), or my favourite, a false discovery rate (FDR) correction using some variant of the Benjamini-Hochberg procedure.
A family of comparisons (hypotheses) is difficult to define, but a loose definition is any group of comparisons in which you expect the difference to be in the same direction. For example, several alternative correlated measures of plant growth would certainly constitute a family of comparisons that should be corrected.
With 25 comparisons, you probably should do some correction. If you're using R, check out the fdrtool or qvalue packages. qvalue is especially easy, because it allows you to input a string of p-values from your 25 comparisons, and returns a list of 25 q-values (significance controlled for FDR).
In any case, make sure you include effect sizes (r, R^2, d). Effect sizes with confidence intervals are much more informative than arbitrary significance cutoffs.