There is a difference between "storing up" treasures and having the necessities on hand to live on when should one become unable to provide for oneself due to illness or old age. Having several fine houses filled with antiques might be thought of as storing up treasures. Having a retirement account available to live off of when one is 80 is another story. To me Jesus was saying not to count on these things as a solution to the big picture issues and not to get so caught up in obtaining the necessities so as to loose sight of our spiritual needs or obligations.
As usual, there are those who take something that was intended as a reminder to be balanced and they go overboard with it. Despite all the evidence around them that we need to be prepared to care for ourselves, they'd have you throw all common sense aside and have you live by the seat of your pants without a back up plan other than hoping God will solve any problem that comes along.