The yarn used in Fetching is Cashmerino Aran by Debbie Bliss. It is a very soft yarn, and knitting it more tightly than recommended will work. There are other Arans, however, where it won't do very wells. The pattern writer acknowledges that this is a tight gauge for Aran yarn: "note that the gauge listed on the yarn label is 18 stitches over 4 inches, while the gauge for this pattern is 21 stitches over 4 inches. That's because smaller-than-recommended needles are used to create the thick, velvety knitted fabric."
The really important thing is to get gauge, or get a gauge that will give you the right measurement, but this means adjusting the pattern. If you're not a very experienced knitter, or don't want to adjust the pattern, then just get gauge. What yarn are you using?
The pattern calls for US size 7 needles, but that's just what the pattern writer needed to get the gauge. You may very well need to go up (the thicker the needle, the fewer sts per inch you'll get) or down (the thinner the needle, the more sts per inch you'll get) to get the right gauge.
If you don't care if the mittens come out as dense as intended, then a good yarn to use would be a light worsted or DK weight-- that yarn will work well with a 21 sts = 4" gauge.