Nope. You can't go into the game expecting to fly the Super-Caps but you can be flying the small ships and flying them well in a very short period of time. Trick to Eve is picking a single path and sticking with it. Jacks-of-all-trades tend to do poorly in that game.
My main in Eve is about to hit 100 Million skill points and there are still ships I can't fly. However, the ones I can fly I fly extremely well.
This is about the best advice anyone could give to someone who hasn't tried eve.
To further refine on it though, you probably don't ever want to fly a super-cap, unless with an alt.
My original main (dxrom) has 68m SP and is geared towards combat and as such has a very strong subcap core as I have never had any interest in putting him into capitals (dreadnaughts, carriers, motherships, titans). My alt-main (Poggz) however is geared towards Combat logistics, she can perfectly (all skills 5) any t2/t1 logistic ship but I have started working towards caps with her, currently she can fly a triage Archon, which is a Carrier and her next step is finishing the gunnery skills for the Revelation which is a Dreadnaught. I've played actively on dxrom since 2009 and created Poggz a year later and trained her for two years before I even started to use her. Granted I could have used her a lot sooner however at the time there were learning skills to train which took time (and I got her to all 5s with them =/) and I had a mix training track with her, first I wanted to use her as a solo pvp pilot with droneboats, then a transport character, then a probing alt and finally settled on being a logi pilot.
Not to dissuade you, but after getting into a ship that you're comfortable flying for the next 4-6 months, you will probably (if you train efficiently) spend that 4-6 months training core fitting and module efficiency skills. Which is where I highly suggest you start (it took me 3 years to finally bite the bullet with dx and do it, but it was worth it) being that these are the skills that will most benefit your ingame ability to properly fit and effectively control any ship you fly in the future.
In my opinion the best route to take on any chr is this:
1. Find a ship you will learn to love and use alot of, generally a cruiser/frigate hull.
2. Train core fitting and efficiency skills. IE: Maxing out your Potential powergrid and cpu thresholds and capacitor recharge rates. Also T2 module skills like local repairs/boosts, damage controls, armor/shield hardeners, etc.
3. Train drones. Drones are awesome and almost every ship uses them, this also opens you up to being able to flying gallente ships which specialize in Hybrid weapons and Drones.
4. Now train into whatever ships you want and their respective weapons.
The reason the order in which skills are trained is so important is because of your attribute points. Perception, Memory, Willpower, Intelligence and Charisma are your stats in EVE and they are at a base of 17 at the minimal, with 14 points left over that can be re-mapped elsewhere. Re-mapping can only be done 1 time a year unless you have bonus remaps available, so deciding what you want to train into by taking into consideration your stat point allocation is an extremely important thing and should be planned out carefully. By default new characters have start with 20/20/20/20/19 for attributes, with their base 1 remap available along with 2 bonus remaps.
It's a lot to take in and might seem daunting and not worthwhile, however there are corps and alliances (guilds essentially) in game that specialize in teaching new players how to play and getting them on track for the things that interest them be it pvp, pve, exploration, marketing, etc. I was personally lucky enough to meet a couple of great guys when I started on the 14 day trial and if it wasn't for seeing them brawl in hurricanes and then help me with the basics, I probably wouldn't even be playing right now. I then moved on to find a corp of still newer players (a good deal of whom I still play with in and out of EVE) and I was able to satiate my dark needs to PK via gate camping with them and performing high sec piracy and warfare.