AFAIK, no.
It depends on the laws in your country. And IANAL.
But, usually, as long as you don't publish (or play publicly), it's fine. The lyrics, the arrangement, as well as the melody, and everything else, are subject to copyright. So if you want to publish a cover, you need permission to use the relevant parts. The usual exceptions of your country's copyright law, like fair use in the usa, apply of course. But I haven't yet seen anything that would allow covers.
Also, from what I've heard of people making covers, they usually first have to look if the original author has allowed covers. Often they allow non-commercial covers only, unless asked. On youtube, copyright is also not handled very strictly, and it has a feature where the original author can claim the copyright offender's revenues, so the video stays online, but the money doesn't (or only partially) goes to the coverer.

Gatze erwache!