I wouldn't say this was "normal" but it is not an unheard of scenario:
You set off with a destination and alternate, both forecasting weather well above minima. When you get close, i.e. commit yourself to landing at one or the other, the weather at *both* goes out of limits.
What to do? Well, you could go off to your alternate, if it looked like it might clear or stay with the destination. If you're at the destination anyway, you get to use your diversion fuel for extra holding, so a better chance of getting in than *having* to make an approach shortly after you reach the alternate.
In the end, if it's low vis., you declare an emergency and land anyway. If there's an ILS, chances are it will work fine, especially if it's CATII or better. The aeroplane doesn't know what the RVR is, so will behave as normal. File the paperwork and carry on.
You don't plan to do this but circumstances sometimes conspire and it's better than flying around in circles and running out of fuel...