Yes, this is very common, though it depends on the type of job and the requirement for your degree for the job. For the current and the upcoming fiscal year, however, there are no new H-1B visa numbers remaining because the annual cap has been reached. An H-1B petition had to be filed in the first week of April. The next opportunity to file an H-1B petition will be on April 1, 2014, unless the employer is exempt from the cap (such as a university). Congress may change the law later this year to increase the annual cap on H-1Bs, so keep up-to-date on what Congress is doing in this regard.
It depends on several important factors including the level of education of the student; availability of a position in specialty occupation; availability of visas for a given year and many others which cannot be assessed on your post which contains no facts
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Technically, the short answer is yes, assuming you continue to maintain your lawful status in the US until the numbers become available again next year. But I agree with my colleagues that it is difficult to determine whether you're eligible for an H-1B because it will depend on your educational background, the requirements for the position, etc.
Yes. However, you should consult with an experienced Immigration Attorney.
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