Amp output versus Speaker imput

If I have an amplifier that is rated at 350 watts output - which I want to keep, does that limit me to speakers that have a recommended amplification of over 350 watts?

I'm in the market for new speakers and there are some I fancy but have a lower recommended amplification that what my amplifier is rated at. Do I discard these speakers or could I play them on my system but not turn up the volume too much. Or given the size of the room concentrate on high amplification speakers. I am budgeting about $16,000 to $20,000 for them.

My listening room is 31' by 10' for 14' then widens out to 15' for the remaining 17'. The speakers are placed in the narrow end of the room. I listen mainly to classical music.

Are there any recommendations that anyone might have on what speakers might be compatible?

Would truly appreciate anyone's input.

Thanks in advance
This is an imperfect analogy, but an ordinary 15 amp wall socket can deliver 1800 watts of power. Yet no one worries about plugging in a clock radio that only needs 25 watts of power.

Speakers are slightly different in that you can turn up the volume control to "push" more power to the speaker than it can handle.

However, that only becomes an issue if you listen at extremely loud volumes with a speaker that is not designed to deliver high volumes.

So question number one would be, how loud do you listen? Many people only listen in the 70 to 90 dB range. We're talking average volume, not the momentary peaks that dynamic music contains. (If you don't have one, spend $50 on a Radio Shack sound level meter. It is very handy for getting a better idea of your listening habits.)

If you are in the above category, even an inefficient speaker will be safe to use with your amp as you'll never be sending anything like 350 watts to it.

If you listen in the 95, 100 dB or higher range and have an inefficient speaker, the level of concern goes up. At that point the mismatch of speaker to your listening habits can increase the chance of incurring damage. However, the same risk is present with a lower power amp when you try to push a speaker to play at a volume level it really wasn't designed for.

Once you've determined that the speaker you buy is easily capable of producing the playback volume you need, I'd cease any further worry regarding the power issue.
No wonder my damn clock radio is so loud!
Almost every case of speaker damage is from too little power from the amp. While it would appear that if the speaker would take 100 watts and the amp was rated at 15 you would be safe, this is not the case. If you overdrive the amp you will drive it into clipping, which will eventually destroy the speaker. The rating is for continuous input, not music. I use speakers rated at, I think, 50 watts and use 300 watt amps often. In a recent test in HIFI News they found that a 500 watt amp worked well with a pair of bookshelf speakers rated at a fraction of that. The truth is these figures more or less meaningless. You can damage your speakers with an amp that is too large, too small or just right if you misuse it, with a little discretion you will be fine. As a classical listener your ears will give up far before the speaker is likely to. The important number is the maximum output of the speaker, if that number is higher than the level you listen at you will be fine.
1 watt of distortion can do more damage than 500 watts of clean power. You can never have too much. Keep in mind though that some of the best sounding amps made are only 3-7 watts.
This is one heck of a pair of speakers which ever they are. I echo the thinking that low power will be likely to do more harm than much more than sufficient power. I would think in terms of something that can comfortably deliver 700-1000 watts.
There are not a lot of amps that do this but McIntosh does have one that makes 2,000 watts. I don't have any idea of what it sounds like or what it costs.
In theory since I have not heard one in action, you can add a Musical Fidelity "supercharger" with either 550 watts each or 750 watts each. I think Audio Advisor has the 550s for a very low $995 close out price. That is down from $1995.00 which was at a heavily discounted price already .

MSRP I think was about $3000. It would appear to be a inexpensive option. It supposely simply adds power only to the signal you deliver claiming to preserve the sonic of your amp.
Musical fidelity products come from either the UK or China . I don't know where they make these. The English MF I am familiar with because I once owed a MF 308 amp. It was a very serious piece of gear that was very solidly built. I only sold it because I needed a more warmish coloration for my analytic, detailed Speakers.
Thank you all for your information. I'm a listener in the 85db class so my choice of speakers has been extended.

Thanks again
Indeed the volume that a large output amp can acheive without distorting is what is critical. Your listening habits are on par with those, who like it a bit higher than the average audiogon member. I gather this from the numerous answers to how loud do you play> think I will post a "how loud" census for update.
Thus changing your speaker to play loudly with less power may be a good idea. BUT I seriously doubt it however. You like these speakers, finding one you like is a tough step to reach. You risk not liking the sound of higher sensitive speakers.
Many of the most sensitive are all range, others seem to have compression horns.* My average preferred level is in the low to mid 90s.
I would not accept defeat. Your budget is ample for used monster amps of very high repute.
Give a look at Higher Fi . com an advertiser here on the gon. They have very impressive high end gear. Much of it within your budget.

P.S.* (I own two pairs of klipsch I almost never play with a sesitivity of my original La Scalas of 104 Db/watt at 4 feet. 104 Db is not a comfortable sound anymore. I have also Heresys at a mere 98 and Hi FI ca. 1960 JBLs 2 pair with the bullet tweeter also compression at 102db. If you use tube amps your system has to have very carefully selected tubes for low noise and you can't have any hum SS or tube. Its all amplified to audible levels with such speakers.) Do I like it well yes I did at one time But not very much lately. I don't play them very often I keep them mainly for sentimental reasons.