Is it possible to eliminate all attachments?
According to Buddhism, attachment(Upadana) is caused by Tanha(craving) and according to the Second Noble Truth, craving is the cause of all suffering. According to the Third Noble Truth there is a cessation of all suffering. According to the Fourth Noble Truth, there is a way to achieve the aforementioned cessation of suffering. 

Hence it is one of the core claims of Buddhism that all attachments can be eliminated. There is no Buddhism without this claim.