What sort of aircraft? The smaller the aircraft, the more likely you are to get in due lower speeds and the ability to avoid cloud without unstablising the approach. The way an operator's manual is worded will be the next thing that affects whether or not an aircraft will make an approach. Some are precluded from making an approach if a go-around is likely, some are allowed to give it a shot no matter what.
We should also consider the navaid giving the guidance. Is it very close to the runway or is it behind you on the approach? In general, given appropriate viz, say in excess of 2,000m then I'd expect to get in most times given cloudbase at minima. But as soon as you put any cloud below minima then the chances of getting in reduce rapidly, like FEW at 600 say 50% chance down to 20% at BKN.
However, if you are asking these questions because you are an airfield operator who would like to save a a few pennies by not installing an ILS, then truly I would forget it. From what I understand, the commercial people in scheduled airlines do not like going to airfields which don't have the full set of navaids or the weather which means they don't need them (I'm excluding the cheap-cheap operators who would land on a road if they could save a few euros).