Yes 
We buy plenty of fresh food (some extra always hanging around), plus drinks, tinned/dried goods, frozen bits, for me, DH and 13mo DS for around £50/week. To be precise, it is up to £55 if I find that I have to top up on milk and bread. The £50 already includes a mammoth amount of whole milk, bread and eggs, between DH and DS alone they get through tonnes of the stuff! Also includes luxury bits (naice tomatoes, proper butter, organic fruit box, nice yoghurts for DS, that sort of thing). We have mostly the essential range and offers and then the other stuff that we like to get that costs a little extra. We shop online at Ocado.
The biggest spend is if you buy formula. When DS was on formula our groceries still cost around the same - he eats quite a bit of food! - but I would have a contingency of an extra £10-15/month (ie about 2 tins' worth) whilst still heavily dependent on formula, particularly when they have a growth spurt.
If you look on the budgeting thread (Juggling January...) there are lots of good tips to keep costs down, for groceries and also other bits. Personally my biggest advice is to always look at the per kg or per gram price, not the pack price, you'd be surprised how much you could save if you're not already doing this. Also be prepared to switch meats/fish and ingredients depending on what is on offer or what is cheaper - eg we have turkey thigh mince, tried it mainly on GP advice, preferred the taste, then realised it's actually cheaper than beef mince too - £6.38/kg, better when on offer, compared with an average minimum of £8/kg for standard beef mince. We eat dark meat when it comes to chicken and shoulder/shoulder steaks for pork. Frozen veg is very handy, usually very fresh, and often works out cheaper.