No. There is really no such thing as negative mass. Even antimatter has mass, which is always a positive (that is, greater-than-zero) quantity.

There is, however, the concept of effective mass or apparent mass, which can be negative. When an object is submerged in water, its apparent weight is reduced by an amount equivalent to the weight of water it displaces. For example, if a 10-pound rock is dropped into a bucket of water and displaces one pound of water, the effective weight of the rock under water is nine pounds.

But what if you submerge something much less dense than a rock -- say, your little brother's head -- in the bucket? Okay, forget that. Let's says a big ball of Styrofoam, instead. (Not as eco-friendly but less violent.)

In that case, the weight of the water displaced by the ball will be far greater than the actual weight of the ball, so when you subtract the weight of the water from the weight of the ball, you'll get a negative number. In other words, the effective weight of the ball under water will be negative, and the ball will tend to rise. It will fight your efforts to submerge it, much as your brother would if you submerged his head (but for reasons other than mere buoyancy).