In the US, there does seem to be a fair amount of competition between the two. I've seen more upscale products that were debuted at Trader Joe's appear on the shelves at an Aldi a month or so later at a slightly lower cost.

On a side note, I had never considered going into an Aldi until I heard a "rally the troops" speech by the CEO of a major consumer products conglomerate. He was worried about how stores like Aldi had upped the quality on their own brand goods and that Aldi's customers were becoming more upscale. He was pushing the conglomerate to promote it's brands and branding to fight off the threat.

The next week I shopped at an Aldi to see what he was talking about and have been occasionally shopping there ever since.