You can have a BSN (Bachelor of Science in Nursing) without being an RN. It would mean you did not take or did not pass the licensing exam (and would be pretty worthless as far as degrees go. You have to have both a degree and the license to work as a nurse).
The title of RN is a legally protected title that delineates someone who has passed the examination to be a registered nurse. There are multiple pathways to being a registered nurse, including not having a BSN, but rather having a diploma (rare anymore) or a two year nursing degree (ASN - Associate of Science in Nursing). So an RN without any further credentials generally indicates an individual who has a two year nursing degree and passed their licensing exam - an ASN, RN or sometimes just shown as RN (Sally Smith, RN). Someone whose name has BSN, RN after the end has a four year nursing degree and passed their licensing exam (Sally Smith, BSN, RN).
The correct format for credentials is to list the highest degree level obtained, followed by the license and then any certifications earned at the end. One can have several nursing degrees but it is correct to only list the highest degree obtained. Therefore, if you have earned a two year nursing degree, then went on to earn a BSN and after that an MSN, you would not be Sally Smith, MSN, BSN, ASN, RN. You would be Sally Smith, MSN, RN.
An ASN degree is called a two year degree, though three and a half years would be more accurate. A BSN is a four year degree plan. Passing the NCLEX (the licensing exam) usually takes place within a few weeks to a couple of months after graduation from school.
A school that advertises its program as RN to BSN is intended for individuals who already have their two year nursing degree and have passed the licensing exam and now want to earn their four year degree. It is not meant for individuals who have no degree or license yet. The course work literally takes you from a two year degree (RN) to a four year degree (BSN). Thus the name RN to BSN.