Yes, neural networks are susceptible to adversarial attacks. No, just adding noise to an image doesn't break neural networks.
In fact, if your technique or model is seriously affected by a little noise this is usually enough to brand it brittle and maybe even a failure, as it's a sign of overfitting. Anyone working in this field knows to look for this and will try to make what they create more robust.
The design of visual captchas is one obvious indication of just how successful AI techniques have been at image recognition in the presence of noise. It's no longer enough to make them a little noisy. In order to resist being solved by mechanical means, visual captchas have to include so much noise that even humans have problems recognizing them.
The issue is that there is no scene understanding. No common sense. No 3D modeling. Just 10x10 pattern matching on a very large fuzzy database of natural images (which works really really well in most cases).
The hype of ML is driven by 3 things: Big companies vying for AI dominance, militaries that want to finally use neural nets that work, and international competition between the West and the East to be the first to largely automate their economies (or AGI if you want to call it that). Catalysts were big data hoarding, GPU training on ImageNet, and then AlphaGo.
What might fool one solution might not fool another, and adversarial examples seem to depend on idiosyncrasies of a particular solution.
The issue is classification currently relies on a very small embedding of the data which is pattern-matched, with no semantics. It has no way of telling that the difference between a dog and an elephant ISN'T that noise gradient, at least some of the time!