I worry (worried) about this, as well. ALOT.
When I first moved to a four-year university (for a dietetics program), the stress caused me to become...Well, depressed. I trucked through the first semester and managed to get through with a 3.0 (decent for someone who could barely get out of bed.) When I decided to transfer to a different college to finish my pre-reqs for an ASN, I was really worried that I wasn't "smart enough" since I suddenly had to work so hard for so little results.
After a while, I realized that most of the depression and lack of motivation came from the fact that I knew that I was in the wrong place. I wanted to go to school for as short amount of time as possible (some of us weren't meant for college - bad to say these days, I know!) but still have a job that I could love. As of now, as I go into another semester of pre-reqs, I am worried, but not as much. I know that nursing is where I need to be from the joy I get from helping others and the interest I have in the field and all that comes with it.
I am positive that nursing school will be a challenge, a BIG one, but I am lucky to be young with few responsibilities and a solid view of where I want to go from here.
All of that said, I truly don't believe that it is an issue of being "smart" enough. It will take intelligence, but I feel most of all it will take tenacity. I think when you begin nursing school things (this is how I feel about myself, anyways) will click and you will become more confident.
For people with anxiety, often confidence is something that comes after the fact. When you get in there, if you really know that is what you want to do, and you start doing well, you will quickly believe that you are smart enough.