Yes. Drowning is a great word to describe it.
There are a lot of really good, qualified grad students who do not get encouragement or reassurance from their advisors. However, I don't think the converse ever happens (i.e., an advisor is confident in a bad student). If your advisor gives you reassurance, att the very least trust your advisor on that!
Echoing what juilletmercredi said, just about everyone feels this at some point. People have different ways of coping. For example, people who appear confident may be totally faking it. Even the person I consider to be THE BEST student in my program--she has a lot of doubts. Even the "weaker" students in the program have them--but by NO MEANS are they bad students. Everyone's pretty good.
You got in for a reason. You're a good student and a good researcher with good potential. I will say, though, don't compare yourself to other people--there are plenty of other grad students who seem perfect, but that doesn't mean you're bad, by any means.
One thing you could do, though, to help convince yourself--examine your performance vs what the advisor expects. Sometimes, advisors are unrealistic. At the same time, some of us could improve. Like, reading an extra paper (or 5, 10) per week, doing a few extra hours in the lab, etc. If you're already at your max, you're probably doing a good job. Even if you aren't, you're still probably doing a good job (unless you are SERIOUSLY slacking, but this doesn't seem to be the case).
Okay, there was a lot of rambling here, but the bottom line is that you're not alone!