Do tyres go 'off'?
Yes! When purchasing new it's ideal if they have a build date within the last 12 months, 12-24months is getting on and if 24months + you better be getting them damn cheap.
If you drive a car (especially noticeable in high performance cars) that has tyres already 6-7 years old even if they've only done 5,000kms, the silica compound just degrades and they'll provide very little grip.
Be wary of tyre shops selling 'cheap' rubber that is already a few years old.