It's a > 100% marginal tax rate. Emphasis on "marginal".
It happens every time the government decides to give something to the poor. A person that just crosses over the criteria used to determine who is poor has a >100% tax rate over that extra income. That means, if they work more they lose money - something called "poverty trap" because it effectively entraps people into poverty.
The solution to avoiding it is to simply give the money for everybody, and create a progressive tax that gets it back from the richer people. Or, in other words, basic income.
A person who receives $100 worth of benefit so long as they make make under $200 faces a harsh cut from $299 to only $200 should their income surpass it. Simply culling overage from the benefit, on the other hand, would mean they would reach a plateau in which they are adjusted to $300 from $200 to $300 worth of raised personal income, which while not be greatly encouraging, would have none of the damning nature of the strict cutoff approach.
Suppose you have a $5k/year benefit at $0 that phases out to $0k/year at $20k. The worst way to do that is obviously to have a cliff: at $20k, you suddenly get $5k/year less, effectively an infinite marginal rate. It's as extreme a trap as you can imagine.
But now consider the opposite, best case scenario that's the softest landing spread out over that entire income range: a benefit that phases out linearly. So at $10k/year, you get a total of $12.5k; at $15k/year, you get a total of $16.25k/year. That means you automatically get a walloping marginal tax rate.
If you wanted to limit benefit phase outs to contribute a maximum of 5% in marginal tax rates, you've got to spread it out over 20x the value of the benefit. A $5k benefit has to phase out over $100k.
Also note that that's by itself: you've also got regular taxes to pay. Worse, few of these welfare benefits have been designed so intelligently, and there are a lot of them, so you constantly hit discontinuities in marginal tax rates that make it really irrational to do any additional work or try to improve yourself.
I hadn't considered the >100% tax rate on the impoverished, either. I wasn't even aware that this happens - it sounds absolutely horrific, and should be abolished posthaste if it's anything like you describe. Can you point me at some reading material on that sort of thing?
As for the $40k-and-above folks, I definitely think that it could encourage exploration by people with little to lose, but I don't think it would significantly affect people with mortgages and families - quitting your job to pursue a dream that fails still means losing your house and not being able to consistently feed your kids even on $10k BI. I definitely think it would help a certain set of people, and I would love to see entrepreneurship encouraged and facilitated; my concern is that it would do so by disproportionately penalizing existing entrepreneurs, by front-loading the majority of the cost for such a system onto them, and making them bear the brunt of the penalty for it if it fails.