Not necessarily. For one, most fMRI studies will have a resting state scan to establish a baseline of what an individual's brain is doing in an MRI setting when it's not really doing anything in particular (typically having the subject stare at a fixation cross and letting their mind freely wander). Then, for the actual task, stimuli are presented and responses are recorded at very specific times with that fixation cross in between with enough time for the brain activity to return to that resting "baseline" before the next stimuli.
There's a lot of signal processing theory, regression analysis, etc. that goes in to it. With that said, the issue that arises is that only the strongest correlates may surface or be observed. There may be a lot of brain activity that overlaps or looks identical between that resting state (baseline) and the stimulation state, but can't really be included because a distinction can't be made, which goes back to your final point. That, however, doesn't mean that there aren't actual subsets of brain activity that we can attribute to a specific activity.
For example, if you're in an fMRI and I show you nothing but a fixation cross, then show you a sad image at a specific time for a specific interval, then went back to the fixation cross, and also showed you a scrambled version of that same image, I can (oversimplified) run a diff check to see what's different between the three. From there, I can remove the overlap between the three from normal baseline activity and activity specifically related to visual processing (scrambled image) to reveal what's specifically different with that sad image. If I see that the amygdala has much higher activity during that sad image compared to the fixation cross and the scrambled image, I can conclude that it has some role in emotional processing (well not really in this example because you'd also need a ton of images, including neutral images, with different orderings of each to really be confident; also really couldn't make that conclusion, just present the correlation because...science).
The point is that there are methods to isolate activity related to a specific activity/stimuli/response/etc. However, it's currently very difficult, if not impossible, to make distinctions in those overlapped areas.
Note: this is obviously an oversimplification of neuroimaging research and analysis