Is there a negative side effect from eating too many fats?

No. Also, when people say "healthy fats" they're usually excluding saturated fat, but it is a healthy fat. The only unhealthy fat is Trans fat, which is basically only man-made (in processed foods). There's a small amount of trans fat in milk, but we shouldn't be drinking milk anyway (calves should).

Make sure the nuts don't have cottonseed oil listed in the ingredients. If you're eating them straight from the shell then you're fine. But you buy them shelled (for instance Planters brand), there might be cottonseed oil. The problem with that is, chances are it's genetically engineered. You do NOT wanna eat any genetically modified foods, ever. The technology is based on outdated, disproven science and is very dangerous.

(Another oil to avoid is Canola oil, but I haven't seen that on any nut labels.)

For nuts, there is a portion size. Factor in the total fat as a variable for the day. So, if you have your nuts, that might be close to the day's limit. Why it is bad or might be bad depends upon your arteries and if they clog as plaque builds up. While this might not be a concern, statistics show that even young people now have cholesterol problems whereas in the mid last century, it was unheard of. Therefore, keep a tally and consider that it is better to err on the side of caution when it comes to fat intake. That's how I do it.

Since nuts have healthy fats and oils, they're actually really good for your body. But a lot of them are high in calories and sodium, which can be unhealthy.

Despite it's PR problem, fat is actually very good for you. You can eat all you want. It's carbs that you should avoid.