How much can I get for an auto accident that wasn’t my fault if I got hit from behind and push to another car?

IF ALL the facts support your very short version of the events, you are likely to be entitled to recover from the at-fault driver.  His/her policy limits are irrelevant except as to evaluating the likelihood of being to actually collect whatever amount you may be awarded but it is totally irrelevant to how much you should be awarded.  That said, it is impossible to give any sort of meaningful answer here for such a question, and it is not even an appropriate question for this sort of forum.  This is for questions as to general legal principles; NOT details on a specific case, let alone any attempt to evaluate the amount of damages reasonably due in a specific case.  The answer is to go to a local attorney to have your case thoroughly evaluated and then valued - much depends upon how badly you were injured; how much it cost for the resulting medical bills; how long it took to recover; if there are any permanent damages; whether you took all reasonably necessary steps to treat your injuries (skip any of the Doc's recommendations? Not schedule follow up appointments as frequently as the Doc recommended or skip some? etc.); whether there were reasonable alternate therapies that could have been cheaper; were there any pre-existing injuries to the same general area; and on and on...See a local attorney to help you through the entire process.  It is what we do.