as above, it's not a linear improvement the bigger you go; it reaches a point where there is a compromise and some things suffer.
things that change, including as discussed above, include:
*tyre profile: bigger wheels = lower profile (to retain same rolling circumference); lower profile = less 'squirm' during cornering, making for better grip; at the expense of ride comfort cos 'less tyre' to absorb bumps
*tyre width: you can usually go for a wider tyre as part of the combo whilst retaining the same rolling circumference; better grip but more noise and more prone to aquaplaning (losing grip over 'sheets' of water) and tramlining (tyres follow the contours in the road instead of where you point them)
*weight: bigger wheel = more metal = increased weight; increased 'unsprung weight' puts more work on the suspension; if suspension not upgraded it can feel 'crashy' and excessively bumpy; reduced acceleration AND deceleration
*TYRE QUALITY: i think this is important and taken for granted; but lower profile tyres tend to be engineered and marketed as higher quality; it's as simple as changing from a lower quality to a higher quality tyre (all else being equal) which can bring significant benefits alone
*4WD looks: it's an optical illusion but the air gap between the top of the tyre and the wheel arch appears bigger and more obvious; this can look ungainly and like a jacked-up 4WD
*cost: initial outlay plus replacement tyres
*higher risk of buckling wheels: less tyre to absorb potholes so more force transmitted to wheel
i agree, that on a car like a 147/156/GT, 17" wheel-tyre combo is probably the best compromise between handling and comfort.
doing a '+1' upgrade (ie: increasing wheel diameter by 1") may not bring an obvious, significant upgrade or difference, especially changing from 15 to 16". a +2 upgrade would be more noticable. once you reach 18" and beyond, you start to compromise on ride comfort, and possibly grip especially on bumpy roads, where a bit of tyre 'give' can aide grip, instead of being skittish; it no doubt remains dependent on tyre design and quality. and once you go to 18s, i think you will want to upgrade the suspension for better damper control, and lowering to reduce the air gap and improve cosmetics.
it's not all bad though, and mostly good. IF you do it right! but that costs $$$; so if you can upgrade the suspension, choose a high quality lightweight wheel (for its given size), and choose a high quality tyre, the results can be excellent.