Earlier quoted context omitted.
Sorry, could you explain what the 10th amendment is? (The Wikipedia page has some assumed knowledge that I do not have as I'm not American.)
The basic reading would be, that unless the Constitution defines a power of the federal government, that power resides with the states and citizens. As mentioned in the Wikipedia article, its addition was basically CYA, as it was and is true regardless of its enshrining within the Constitution. As such, its existence does not really mean anything beyond what the Constitution already meant. It just provides a handy sh…
There are limits to this, some of which are set out in South Dakota v. Dole, 483 U.S. 203 (1987). These limits (quoting wikipedia) are:
* The spending must promote "the general welfare."
* The condition must be unambiguous.
* The condition should relate "to the federal interest in particular national projects or programs."
* The condition imposed on the states must not, in itself, be unconstitutional.
* The condition must not be coercive.
Especially the last condition is relevant. It means the withdrawal of funding cannot be so harsh as to be clearly funding. I believe the actual wording used is that the threat of withdrawal cannot be a 'gun to the head' of the states.