Correct. Though once you reach a certain age, it's up to 7k.
You probably meant to say "stay on the programs" rather than "get off of programs". Because that's a huge differentiator that most financial planning and advice doesn't address.
In NY qualifying for Medicaid or Essential Plan is huge. If a person also qualifies for childcare vouchers, that enormous. Section 8 (or other subsidized housing) I don't even know how much (nor do I know if a 401k might disqualify one or significantly reduce the subsidy).
All of that being said, accumulating too large of a balance in a pre-tax retirement account can create other problems down the road, as moving money out of the plan is a taxable event (taxed at ordinary income rates, and a penalty if dones too early).
The goal is to be able to live without programs without ever being in the sweet (sour) spot where every dollar in income results in more than $1 lost in programs.
For someone making the perfect amount of money to maximize programs they still pay 7.65% FICA and lose:
EITC 21.06%
Food Stamps 30%
Section 8 33%
This is already over 90% of the additional income, I may have missed some and this doesn't include other smaller programs like LI-HEAP. By the time you get done you're literally coming home with less money, the goal of putting it into a 401k or IRA is to launch you to the point where you are ineligible for at least some of the programs so you no longer lose more than you make.
The ramifications down the road will be significantly smaller even for someone who never gets off programs because they receive less from programs by the time they reach 70 1/2. For example EITC is tiny without eligible children.