Before revision:  In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 600{599}  v\geq \mathbf 1. Using ideas similar to our vector bin packing result, we show that for vector bin covering there is no APTAS for dimensions greater than or equal to 2.
After revision:  In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than 600{599} ^d, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin B, we have \sum_{v\in Bv\geq \mathbf 1. Using ideas similar to our vector bin packing result, we show that for vector bin covering there is no APTAS for dimensions greater than or equal to 2.