Has anybody personal experience in trying this ? Is it not natural to have some attachment? I realize attachment is a cause of suffering but is suffering not natural?
Attachment to that which is impermanent will result in suffering. Attachment is clinging or holding onto something, resisting change, seeking to duplicate past experiences when that is simply unreliable or even impossible.
One can live a normal life without attaching - we can do this by ceasing to fall prey to our habitual judgements and reactions, cultivating a mindfulness that sees things as they are, without a faulty bias or colored glass to see with. When we are not mindful of our feeling, perceptions, or judgements, we get carried away with the proliferation resulting from that. One can enjoy music or a garden without becoming attached to it - you know these flowers will wither, and the music will fade to silence, yet they are still pleasant. As long as you are aware of their Impermanence and the fact you judge them as pleasant, you'll be fine.
Seeing clearly, without judgements, leads to Equanimity, which leads to non-attachment, which leads to peace, which is freedom from suffering. The more Equanimity, the less suffering. When we see clearly, we see all things as impermanent, not capable of satisfying our desire, and therefore not worth attaching to or identifying with. When one truly knows attachment as the precursor of suffering, it is easy to avoid, like you avoid putting your hand in open flame.
When attachment arises, become mindful of it. Ask, is it skillful, leading towards relinquishment and freedom? Ask, is this attachment permanent, satisfying, or truly mine? Then, simply observe as you let it go. How does it feel to release an attachment, to watch a desire fade away? When you resist the urge to eat that candy bar, how do you feel? Try it out. Come and see. ;)
I think it's important to define what do you mean by "natural".
Are you defining it as "statistically normal/frequent", or maybe as "intrinsic and necessary to the human condition"?
If it's the former, then I'd argue than a lot behaviors are normal, but that does not make them wholesome, beneficial nor necessary; if it's the latter, then I'd ask you: how do you know that suffering is necessary and inseparable from the human condition? Is it because most people say so? Is it because you've never seen someone that could've gone beyond suffering?
And from those last questions, I'd ask: How do you define "suffering"?
Remember that the Buddha used the word "dukkha" to describe the mental phenomenon he wanted (and, in my opinion, successfully did) to eradicate through the Path he (re)discovered: the Noble Eightfold Path.
"Dukkha" is polisemic word, that is, it has multiple possible renderings: sure, one of them might be "suffering", but such translation is too broad and culturally loaded. Some translators prefer instead "stress" or "pain". I think the most optimal term might be "unsatisfactoriness".
And thus, the Noble Eightfold Path is the training to uproot insaisfaction, through the eradication of its conditions; namely, craving/greed (tanha/raga), aversion (dosa) and ignorance (avijja/moha).
By getting rid of such conditions, insatisfaction cease to be present. By reaching this state, the person is at peace, no matter what outer conditions may arise or cease. In other words, such peace is unconditioned, and not dependent on having this or that. Physical pain may still arise, but the mind becomes unshakable and not influenced by it.