I don't see any reason why a similar cryptocurrency, or even another blockchain (am I talking out of my ass here?) couldn't compete against the current Bitcoin blockchain. Back before legal tender laws and the creation of the central bank, private mints were in competition with each other. Perhaps another system has some features Bitcoin doesn't, or for whatever reason gains hold of a niche. This is not a bad thing in any case. Especially if any competing currencies have decentralized transactions and non reversibility like Bitcoin, trading would be extremely simple. Perhaps even completely transparent. They could easily exist side by side.
My opinion: adding another blockchain (if it was possible) would lower the total value of BTC IMHO.
There are going to be 21 Million BTC in total, since we have a "fixed" quantity of money, inflation is kept under control. As other said, it's just like gold, there's a scarce quantity out there.
Then someone comes in and opens another blockchain/cryptocurrency. Since for the end user paying with BTC or "new bitcoin" (hereafter NBC) will be absolutely indifferent, this will mean that suddently a huge mine of gold is found somewhere. Newcomers and old miners will start harvesting the NBC and when enough NBC will be around, BTC value will drop. If the end user is given the opportunity to use both NBC or BTC, it will take time for both of them to lever, but the ones already owning some BTC will probably suffer a lot from this situation.
Speaking of scarce resource, I guess that at worst, 0.0000001 bitcoins (7nth decimal point) are gonna be worth one candy. Since I can buy candies at 5€cents, I expect 1BTC not to rise at more than 500'000$ in value
This reminds me of a story on a comix, with the main characted being sent to a parallel dimension in which 1£ was worth so much they had to pay a coffee with μ£s (micropounds).
Let's speak also a bit about psychology: normal people tend to think at a small number as a small value. For this reason I don't actually expect people to accept buying bitcoins if their price gets too high. I guess a show stopper will be when 1 BTC will be worth more or less a dinner at the restaurant. This realistically means we are never going to see people get bitcoins for more than 20/30$ each: the only way out is to give 0.001 BTC a brand new name, maybe "SBC", small bit coins.