Both are true, it depends on what the income is at, and what is impacted.
If you are at the edge of the 25% federal tax bracket, and if you make a lot of capital gains, it would probably be really good to get down to the 15% bracket so that long term capital gains and dividends taxes are reduced.
In other cases, you're not considering a tax bracket. For instance, say there is some benefit that is no longer accessible if your income is above a certain level. As soon as your dollars go above that limit, the benefit is gone. Just like above, these are situations where the cutoff is hard set and once you are over the cliff it is gone.
The other side is that the US uses a bracketed tax system so that each additional dollar has a certain marginal tax rate. For this case, each additional dollar is only taxed a certain amount.
If a person makes money, they need to be mindful of all of these things, see how low they can get their taxable income, what tax-advantaged accounts they can divert income toward to get the most benefit, and whether that would be advantageous based on the benefits they gain vs having the money now.
So it is all true, and it is best determined with a comprehensive graph that accounts the cliffs and the tax brackets.
In direct answer to the question you posed though, if a person makes 250k in one year, and then 0 the second year, it would be in his best interest if he could find some way to take those top dollars from year one that are taxed at a very high marginal tax rate, and push them off to year 2 where they would be starting from the bottom of the stack.
Naturally, if he simply made the same or more money the next year and every year after that (inflation-adjusted), he might find out that it was not any better to try to displace the money to a later year, because he could never get his tax bracket to a lower level.
On this forum, if a person is trying to FIRE, they might have some years of low income after RE where they can start claiming that 401k / IRA money at a lowered tax bracket.