A bivi can be really good - but you have to be prepared for short, uncomfortable and sometimes chilly nights. I've spent several nights out in my bivi and am using it increasingly - last night I camped just beneath the summit of Skiddaw and rose at half five to be on the top, in cloud, by 6. I did the Cumbria Way in one in three days last year, and can confirm they can be horrific in the rain. There is a weight saving over a tent if you go for a simple one, like my Alpkit Hunka (420g) and if used with a lightweight sleeping bag (my ME Xero MM and balloon bed weighs under 500g). I now carry all my overnight gear in a 20L pack, even for 3 or 4 day trips. If you want to go ultralight, don't mind discomfort and are operating on a budget - the Hunka's £30 and there are a couple of Rab bags for about £60 - it's a good option. The budget was the main reason I bought mine over a tent! 

Defineatly a bivy bag is a bad idea for such a long time. However your happier for something a bit bigger, don't want to spend lots of money, and still keep some gear inside and dry, you could go for the Coleman Avior. It's not the best quality, but it all depends on how much you want to spend, quite a cheap and easy way to start things off with though...

A bivy bag would be a poor option this summer,much better to go for a lightweight tent,after all we are doing LDP' for enjoyment not an endurance test.

.....but there's no real weight saving over a proper lightweight tent.

Indeed! Why make yourself miserable when you could have the luxury of a tent for about the same weight. Okay, you might be lucky and get wall to wall sunshine for two weeks, but I doubt that much. As the folk above have said, bivvies are fine for a 1 or 2 night trip, but as soon as it starts raining they're not so much fun any more.

A bivy bag is good for maybe two nights in our fickle weather. Cooking is always outside remember, and all your gear is coming in beside you and if it's wet it's going to be misery after a while.

Bivies are great for quickies (I did the West Highland Way with one earlier in the year), but there's no real weight saving over a proper lightweight tent.