Lack of Jewish Ancestry?

The question was,

As an Israeli, if a DNA test was done that concluded completely that you had no relation whatsoever to any Jews and your family must have converted in Europe, would you pack your bags today?

This theoretically could happen but it would be very rare. It may be that the founder of a family was a convert but after a few generations they would have intermarried with Jews who had always been Jewish.
Actually I once met a great and well-known Rabbi who told me that on the gravestone of his earliest known ancestor (in Europe) was inscribed "Avraham Ha-Ger." This would usually be taken to mean 'Abraham the Convert' though other explanations are possible. It did not seem to worry him one way or the other though I assume he would not have wanted it widely publicized either. These days I doubt that they would write such a thing but maybe then it was different. Suppose theoretically that a situation arose such as outlined in the query? Why should it matter? Everyone comes from somewhere. Converts are like migrants. People migrate. Where did the citizens of the USA come from? Migrants show courage and initiative and individualism. Converts join the Jewish people and come to Israel every day. It takes guts to do something like that. Some converts may have Jewish ancestry but others assumedly do not. They live here. They like it here, on the whole. Some of them have problems adjusting and leave. Such is life. The Jewish people is an organic body. It can accept citizens. Once accepted they have the same rights as everybody else. The Land was promised to the Israelite Nation. In order to confirm the promise they should keep the Torah but that is a matter between them and the ONE who gave the Torah.

Zechariah 14 
So the angel that communed with me said unto me, Cry thou, saying, Thus saith the LORD of the hosts; I am jealous for Jerusalem and for Zion with a great jealousy. 15 And I am very sore displeased with the Gentiles that are at ease; for I was but a little displeased, and they helped forward the affliction. 16Therefore thus hath the LORD said, I am returned to Jerusalem with mercies; my house shall be built in her, saith the LORD of the hosts, and a line shall be stretched forth upon Jerusalem. 17 Cry yet, saying, Thus saith the LORD of the hosts, My cities through abundance of good shall yet be widened; and the LORD shall yet comfort Zion and shall yet choose Jerusalem.

The Israelite nation from the beginning accepted foreigners who identified with them.

Ruth (NASB) 1: 16 But Ruth said, 'Do not urge me to leave you, or turn back from following you; for where you go, I will go, and where you lodge, I will lodge. Your people shall be my people, and your God, my God. 17 Where you die, I will die, and there I will be buried. Thus may the LORD do to me, and worse, if anything butdeath parts you and me.'

Judges (NASB) 2:
 12 '˜Now therefore, please swear to me by the LORD, since I have dealt kindly with you, that you also will deal kindly with my father's household, and give me a pledge of truth, 13 and spare my father and my mother and my brothers and my sisters, with all who belong to them, and deliver our lives from death.'14 So the men said to her, '˜Our life for yours if you do not tell this business of ours; and it shall come about when the LORD gives us the land that we will deal kindly and faithfully with you.'