There are three main types of DNA tests. The replies so far have dealt mainly with autosomal, which is good for about 3-4 generations back. After that, the signal-to-noise ratio in the test becomes problematic. Also, Ashkenazim have a high incidence of endogamous DNA. That contributes to frequent false positives when the autosomal test attempts to go back further than about 3 generations. The response from "Sally B" above gave the correct math and referenced this limitation associated with endogamy. Test agencies are trying to sell DNA kits, so their marketing literature is not likely to feature these autosomal test limitations.
Mitochondrial DNA is suitable for tracing the female line. It goes back many generations. I do not have experience with this test, but I expect it is capable of going back further than any Jewish genealogy you might have. However, most Jewish genealogy follows the patriarchal names. So, mitochondrial DNA research (the matriarchal line, mother-to-daughter, mother-to-daughter) has limited utility. Not being sexist. Just the way it is. Blame our ancestors and their patriarchal proclivity.
A reply from "vivs" above mentioned Y-DNA. This traces the patriarchal line, which is father-to-son, father-to-son, and so one. As long as the direct patriarchal line is not interrupted with a daughter (which breaks the Y-DNA transfer to the male in the next generation), this DNA can prove a common male ancestor existed going back thousands of years. It is easily capable of confirming a genetic relationship among males before the end of the patronymic period (circa 1750-1800). Again, the limitation will be the genealogy, as precious few Jewish genealogical records reach into the patronymic period. You may uncover conclusive proof of a close genetic relationship between two males using Y-DNA, but the genealogy may not be sufficient to tell you how or exactly when those two males had a common male ancestor. You probably will never learn the name of that common male ancestor, if you have to go into patronymics.
You asked about Portugal vs Ashkenazim. Portugal and Spain are usually Sephardic. There are Y-DNA markers that are associated with Sephardic ancestry, but I do not have sufficient knowledge to confidently explain it. I am participating in research that involves 150 Jews with oral history of being Ashkenazim and cohens, who have been proven by Y-DNA to descend from a cohanim line that started with an unidentified common male ancestor in biblical Israel, circa 500 BCE. This research is being led by serious geneticists who make me a novice. There is tantalizing evidence of a Sephardic connection in some of this DNA, despite the overwhelming preponderance of Ashkenazic ancestry among these 150 participants. A plausible explanation is a Sephardic line of cohanim left Spain/Portugal in circa 1492 or thereafter, intermarried with Eastern European Jews who were probably also cohanim, and became comingled with Ashkenazic DNA. If it can happen in this DNA study, it can happen in your situation, too.
Ken Domeshek. (FTDNA project administrator, but not representing or compensated by Family Tree DNA)