The Buddhist notion of non-attachment is a basic principle of belief. If I understand it correctly (and it is possible I don't) it calls upon the individual to detach from material possessions, striving for acclaim and fame, and not being concerned with success or failure.
There is also an understanding that attachments to other individuals can only bring ultimate suffering. Life is temporary. Everyone we know and love will die and we will suffer deeply because of this. Non-attachment offers us an opportunity to move through this lifetime with a pleasant sense of happiness, yet without the extremes of joy and deep sadness which afflicts most mortal beings.
This attitude seems quite plausible if one considers the monastic life to be the preferred one. Non-attachment is much easier and understandable when one has no spouse or offspring. Unfortunately for most of us this is not how we live our lives.
We cannot be other than attached to those we love. To attempt to hold back our feelings and connections is to be other than the human beings that we are.
Now I do agree that obsessive connections can occur with loved-ones. Many people "live through" the lives of their children, spouses or parents. This is clearly not a desirable situation. Allowing our loved ones to make their own choices (which they will do regardless of our opinions) is part of their own soul's journey, their own karmic path.
Clearly possessive attachment to material goods is often tied to un fulfillment and despair as well. Competition to have the biggest, most expensive, most opulent lifestyle is clearly absurd and irrational.
Yet to us "householders" attachment is who we are. If understood in terms of balance–of expressions of love/concern with recognition of individual choices–than it will continue to be an essential part of our lives.