In short, no, but there are things to watch. It can't be too strong for it really, per se, but mixing newer and older hardware will always require watching out for something. ATX 2.x+ vs ATX 1.3 is the only problem that comes to mind on this topic. Putting a strong +12V PSU on a Pentium III system it "should easily be able to power" could be bad since modern PSUs emphasize the +12V rail and Pentium III systems utilized the +5V rail more, but that's the only real example of where I could think putting a strong newer PSU on older hardware could be bad, and I doubt that's your case, so I'm going to say no. If it's both an ATX 2.x motherboard and an ATX 2.x PSU (pretty sure the PSU is given the wattage), it will be fine.
In short, no, but there are things to watch. It can't be too strong for it really, per se, but mixing newer and older hardware will always require watching out for something. ATX 2.0+ vs ATX 1.3 is the only problem that comes to mind on this topic. Putting a strong +12V PSU on a Pentium III system it "should easily be able to power" could be bad since modern PSUs emphasize the +12V rail and Pentium III systems utilized the +5V rail more, but that's the only real example of where I could think putting a strong newer PSU on older hardware could be bad, and I doubt that's your case, so I'm going to say no. If it's both an ATX 2.x motherboard and an ATX 2.x PSU (pretty sure the PSU is given the wattage), it will be fine.
Edit: What do you need a 1Kw PSU for? Unless you're planning on getting a very power hungry CPU, maximum amount of RAM modules, four or more hard disks, and a two or three very high end GPU setup, you don't need it. You'd be far better off getting a lower wattage PSU. Wattage does not equal efficiency or quality. Most standard enthusiast systems (a dual or quad core CPU, two or maybe four RAM modules, one or two hard disks, and higher-end [not extreme high end] GPU) don't need more than a good 500W PSU at most (and yes, that takes into account leaving extra room, obviously). I'd even say my 610W PSU is probably a little overkill for me, and I have four RAM modules, and four hard disks.