I have done so on many occasions as well as used contractors from consulting firms. I have worked as a consultant (wireless communications engineering technology) for many years.
You are basically an IRS Form 1099 contractor for tax purposes
unless the consulting firm actually hires you as an employee (W-2 tax status). Both have implications that you should understand beforehand. For example, Form 1099 personnel are not eligible for company benefits, healthcare, etc., hence your agreed upon fees as a 1099 contractor to the consulting company should take into account your need to file estimated quarterly tax returns, pay for your own health insurance, etc.
In short, when seeking positions at consulting companies, your first question should be "Is my status 1099 or W-2?"
As a contractor at another firm you must make sure you are not treated as an employee.
Microsoft got into plenty of IRS trouble doing so and ended up having to pay certain employee benefits to contractor personnel. Review the contractor agreement to make sure you understand what is expected. The agreement will also include provisions preventing or laying down restrictions for the company you are working at from offering you a full-time job.
Contractors are also excluded from many functions of the company they are working with at the time, e.g., some meetings, company functions, etc. Try not to take this personal.
Basically you are a seen by the company you are sent to as a "hired gun" with the expectation that you are able to hit the ground running in your specialty area without a lot of hand-holding.
Whatever you do, never get into water cooler discussions with the company's regular employees about your pay, billable rates of your consulting company, how messed up things are at this company, and so on. You represent the consulting company that sent you. Maintain high standards.
AMR