Every place is different but ya, you could end up taking all those classes and do great in them and not get into the nursing program
There is always risks in life.
But really there is that chance with *any* professional program that people have to apply or compete to get into. I have changed to pre-med (my major is Biomed) and for the med school we have 5,000+ people applying for 120 spots, PA school its around 1-2000 for 50 spots, dental school is about the same as med school, dental hygene is 500 for 30 spots, etc
When I was doing nursing I looked around at all the schools in my area and took all the classes so that I would have the option of applying to any of them (since they all required different classes, tests, etc) so make sure you check out all the different schools. If you can get a job as a CNA or at least volunteer in a hospital or nursing home.....some programs count that and will give preference. You could also start off as an LPN and then bridge over to RN/BSN.
Just do well in your pre-reqs....I would really really try to get an A in everyone of them and get some experience in the health care field and you will probably do just fine!