As Art notes, rim width is a problem. Most tire shops will refuse to mount tires on rims that are too narrow, but I found one that would; ran 185/65-15 on stock TR3 rims for a while and they worked out well for me. Steel TR6 rims are 5.5" wide and fit great with 185/65; but any wider than that can be a problem.
And of course you have no hope of getting wider tires into the spare tire compartment. If your car was made before TS60000, even 165 tires won't go in easily. I used to keep a rope tied around the spare, just so I could get it out.