Receiving an offer letter for a new job is usually a sign that the employer intends to give the job to you. However, it's not an employment contract and can be rescinded at any time. If that happens to you, you can apply for unemployment benefits in your state, but you must meet the same eligibility requirements that any other claimant does. Specifically, the eligible job separation requirement may pose some problems.
Some companies provide offer letters for their potential employees, which includes company information, the position and salary information and the deadline you have to accept the job by to receive these terms. While offer letters can provide piece of mind to potential landlords and other creditors, it's not an employment contract. Either party generally has the right to refuse the offer for whatever reason they'd like. For example, if your background check comes back with certain blemishes, the company may rescind your offer letter. On the other hand, you may get a better offer elsewhere and not take the offer presented to you.
When you apply for unemployment benefits, a major eligibility requirement is that your job separation be through no fault of your own. If your employer terminated the relationship, it can't be for just cause, which would be reasons attributed to you and your behavior. If you voluntarily left, you must have just cause, which would be reasons attributed to your employer or his actions. To verify the reason, your state contacts your former employer to check on the reason for the job separation.
If you find yourself unemployed after the employer you were expecting to start working for rescinds his job offer, the question falls to who was your last employer. Laws vary depending on the state, but generally it's the last one that paid you a salary. (In some states, you must have worked a certain amount of hours.) So if that potential job never actually paid you, your unemployment eligibility is based on the last job you had. If you left voluntarily to go to the new job, you don't qualify for benefits.
Although unemployment insurance programs have some federal guidelines, each state makes its own unemployment insurance laws for the most part. So whether your last relevant job was the last one that paid you a salary or the last one you worked a certain amount of hours at can vary. Without knowing your state of residence, it's difficult to be specific about these things. For information about your state's specific unemployment insurance laws or your individual claim, contact your state's labor office.