Islam does not forbid saving up wealth in all cases, rather what is forbidden and what a stern warning is issued against is not paying zakaah on one's wealth. But if a person pays zakaah, he is not included in the condemnation.
Allaah says (interpretation of the meaning):
"And those who hoard up gold and silver (the Zakaah of which has not been paid) and spend them not in the way of Allaah, announce unto them a painful torment" [al-Tawbah 9:34]
Abu Dawood (1564) narrated from Umm Salamah (may Allah be pleased with him) that the Prophet (peace and blessings of Allah be upon him) said: "That which reaches the level where zakaah is to be paid and its zakaah is paid, is not kanz (hoarded wealth)."
Thus it is clear that Islam does not forbid storing up wealth, rather it forbids not paying zakaah on it.