The contracting company takes a significant cut of that.

For hard numbers, 80% markup of wages paid out is typical (at least for large contingency recruiters and IT contractors), though it can be higher.

$250/hr billed usually results in $125-140/hr in the contractor's pocket, and $110-125/hr for the place that sent them out.

In my experience its about 30%(ish) to the worker gross, which means that the worker gets about 20% after tax. It can be slightly higher or lower depending on the situation.

Depends on the org, and the rate. For high rates for specialists (well in excess of $250/hr), this is true (contractor sees 30-40%). For lower-end shops and lower rates (think your Robert Half, your Randstads), the contractor tends to take 50-60% of the bill rate.

My sources are VPs from some of the large contingency recruiters, founders of some mid-range consulting shops (~500 person headcount), and having to help run a smaller consulting shop (~20 employees) in the past before going fully independent.

You are assuming 100% utilization, which is almost never true, and even if it were, most of that money goes to the contractor's employer.

I pulled $250/hour out of my ass.

As for the rest, I am not talking about consulting-style contracts where a company is billing out people individually. I'm talking about contracts for development of bespoke systems, provide specific services, and so on.

When Raytheon gets a contract, they get it to build X. They need to account for the work, and the amount they get paid for labor is the number of recorded hours times the agreed-upon rate for that specific type of labor. There is frequently also a soft cap on the number of hours that can be billed in specific periods. The billed rate covers not only the employee salary but also benefits, overhead, and profit.

Depends, for a leader in their field that's on the low end. But, I understood $250/hr to be what the firm received -- usually contractors don't get nearly that much, but I don't have any experience with government contractors.

From my experience, you're dead on for the majority of "staff augmentation" full time contracts that come from tech-related consulting+contracting firms. Most organizations I've been around (East Coast, DC region) charge around $250/hr to big government, and the people they assign make anywhere from $40-$125/hr depending on the specifics of their role.