jQuery has a considerable performance cost for the abstractions it provides. It doesn't just call the native methods under the hood, it has to provide additional abstractions like chaining by wrapping every returned object into a jQuery object. In a lot of cases these abstractions aren't helpful because you need to understand exactly what jQuery is doing. If you compare the cost of native methods over what jQuery does you easily add 500 lines of javascript code on every method call.

Additionally, jQuery may not provide the abstractions you need or want. If you are using a high level framework that manages templating I would argue that you should use the abstractions designed by the framework and not the one-size-fits all ones that jQuery provides. There is a lot of functionality that jQuery provides that may not be relevant at all for you. jQuery animations are one example.

Using jQuery doesn't stop you from running into browser bugs. jQuery could help you out with basic things like selection and event handling but modern browsers are already much improved in this area.

I very much dislike the spaghetti code that over reliance on jQuery plugins results in. You have pieces of code that rely entirely on jQuery and can't be refactored out or reasoned with clearly. It is terribly unidiomatic in any case.

EDIT: That being said jQuery does have its place.