Partial page loading and sites not accessible generally are a pointer either (as Milo says) that your router is on the way out, but can also be:
- the wrong MTU setting on the router (this dictates the packet fragmentation size and can result in retransmits and loss if incorrectly set)
- a routing issue further in the internet
I would suggest, before you replace your router or look at the MTU issue, you ring up your ISP and explain the problem to them, as I saw this issue on a customer here who's ISP had recently initiated an IPv6 programme on their network and had made a config error. (Kingston comms for those of you in the UK)