I know from my time working at an ISP that possession of three or more pictures showing someone under 18 in a sexually suggestive pose, or a picture that focuses on their genitals, regardless if they were nude or not, would trigger the Law. i would anticipate that having more than a few such pictures in your browser cache would indicate to a Judge that you intentionally sought out the material.
But, as with all things, it would depend on the Judge. Some can be rational, some are rabid about such things (I had heard of some Judges throwing nudist parents in prison for having vacation photos of their kids at the beach).
Obviously it's a case of C.Y.A. If you hit a website that has it, immediately disconnect and inform the authorities what you discovered. That you 'did the right thing' would go a long way toward convincing a Judge that it was unintentional.
Edit:
BTW - what makes you think the NY or Fed .gov isn't already monitoring your internet traffic? It may not be real-time, but I garauntee that your ISP has devices in place that can track your history.