Is it possible to remove only some cartilage from the nose with touching the bone? (Photo)

Hello and thank you for your question.  You appear to have thick nasal tip skin.  It is possible to work on the cartilage without performing osteotomies.   Based on your
photograph, you may benefit from a tip refining rhinoplasty. Your surgeon can
accomplish this by trimming, suturing, and reshaping the cartilage in your
lower nose.   Make sure you specifically look at before and after
pictures of real patients who have had this surgery performed by your surgeon
and not just a computer animation system. The most important
aspect is to find a surgeon you are comfortable with. I recommend that you seek
consultation with a qualified board-certified plastic surgeon who can evaluate
you in person.
Best wishes and good luck.
Richard G. Reish, M.D.
Harvard-trained plastic surgeon

Whether or not you need the bones of the nose altered depends on the changes you are looking for.  The nasal tip is composed of cartilage, while the upper portion of your nose is composed of bone.  If you are looking to only modify the tip, this can often be done with alteration of only the cartilage and no changes to the bone.  If you have a bump or the upper nose is too wide, then this will require adjustments of the bone as well.  It is best to discuss this with a rhinoplasty surgeon so that you can get a good idea of what is needed to achieve the look you want.  

Improving the nostril width and refining a flaring tip to achieve an improved natural appearing cosmetic result are frequent requests of patients wanting nasal refinement. It is possible to improve the cosmetic appearance of the tip alone without altering the rest of the nose in some patients. Whether the optimal option in your particular situation would be a tip rhinoplasty alone or a full rhinoplasty would best be determined at a consultation. Either procedure can be performed with a closed or open approach.

Following the advice from a surgeon on this or any other website who proposes to tell you exactly what to do based on limited 2 dimensional photo without examining you, physically feeling the tissue, assessing your desired outcome, taking a full medical history, and discussing the pros and cons of the operative procedure would not be in your best interest. I would suggest you find a plastic surgeon with extensive nasal surgery experience who is certified by the American Board of Plastic Surgery and ideally a member of the American Society for Aesthetic Plastic Surgery (ASAPS) or a facial plastic surgeon (ENT) that you trust and are comfortable with. You should discuss your concerns with that surgeon in person. 

Robert Singer, MD FACS

La Jolla,California