Sure. The way the H1B works is that it is tied to your employer. The employer knows this. You know it. This creates leverage on the part of the employer. How? Two ways: One, you get fired, you are out of status immediately, you have to leave the country ASAP or find a job immediately. So, you make sure you don't get fired. Two, an H1B employee is supposed to be paid the prevailing job market rate. For CA, San Francisco County, the max is around 100K. It is a pretty sum. BUT. It is also less than market rate as of six months ago (when I was job hunting). Now, most big companies don't need to exploit you. But a wily manager can easily fuck with you with enough awareness of the issue.

Personally, I have noticed this phenomenon at smaller companies where a H1B candidate is potentially more attractive. I got a green card very recently, however, I am obviously not an American and am relatively young enough that I can be suspected of not having a Green Card. I have had a few situations where hiring managers were rather disappointed that I didn't need their help to acquire an H1B or for them to file for my green card.