That depends on what you mean by "poverty" I guess.
Wealth is, of course, the measure of value. Poverty is sometimes referred to as the lack of value (most famously in Matthew 5: "Blessed are the poor in spirit."). In that sense, no. Absolute wealth - abundance of value would eliminate poverty.
However usually poverty refers to absence of money or monetary worth. In that sense, poverty (lack of money) and wealth (presence of [monetary] value) don't have much in common. I.e. a person can be poor and yet still be wealthy. E.g. An person in America could be homeless, and yet own expensive real estate in Europe. She would be poor and wealthy.
Is it possible to eliminate poverty and still have rich people? (This might have been the question you intended.) Yes. In the United States, for example, nobody is poor (compared with 10th century Bantu people) and yet there are rich people. But even if the lower classes in America live better than 15th century princes in Europe, they might still see themselves as poor.
I hope that was a thorough enough answer. If anything needs clarification, let me know
