"Allowing" you to work for 4 months is legally irrelevant: unless you had a written employment contract which limited the reasons or way you could be fired, you were an employee at will and could be terminated at any time for any reason whatsover, including a delayed background check. (Also, look at this way: you got to get 4 months of pay which you might not otherwise have received.)The only real issue is whether you had an enforceable contract. An enforceable employment contract must be in writing and must place limitations on termination--whether by limiting the reasons you could be terminated, or requiring a certain process, or stating you are employed for a certain length/period of time (unless you commit an act which allows termination "for cause," like violating supervisor instructions or company policy). If you have a contract limiting how or why you could be terminated and you were terminated in violation of its terms, you could file a "breach of contract" lawsuit to seek compensation for the wrongful termination. But without a written contract like the one described above, you could be terminated for this reason, even after 4 months.