Yes, you could be responsible for any missed assessments. Before you put in an offer, your agent needs to call the buildings management company, ask to speak with the property manager for this particular property, and find out if any of the taxes and assessments are passed due. You want to make sure the taxes are up to date, and they should also be able to tell you if there are any missed assessments and if so, how much you'll have to pay. There will probably be some late fees that apply as well, so ask for those too.
Unfortunately the PAL (paid assessment letter) usually comes out late in the attorney review process, in a competitive bidding, multi-offer situation buyers often have to assume the worst when they make their offer. At least that way you could be pleasantly surprised!
Hope this helps,
As always discuss with your real estate agent. What has he/she advised?