However, don't panic if it doesn't. Your orchid might be taking the time to grow some new foliage. If it's working on growing leaves, it probably isn't going to put much energy into spiking and flowering. Or it might spend some time building its root system. Basically, as long as the leaves are green, you've got a happy plant. My sister was baffled because she trimmed her spikes but her orchid appeared to not be doing anything. I just told her to keep watering it normally and make sure it's getting enough light. After about three months, just as she was getting ready to give it to me to see if I could get it to sprout, it suddenly shot out a spike from the top node. Apparently it was busy doing other things for three months. Anyway, patience is key. As long as the plant is looking good, keep doing what you're doing.
Sometimes the spikes you snipped will turn yellow all the way to the base a couple of weeks after you snip them. That's okay too...it just means the orchid decided it was done flowering for a little bit. If the spikes turn yellow all the way to the base, snip them off as close as you can to the base of the plant, then give the orchid some time to "rest." Maybe move it a little further from the light and cut back on the fertilizer for a couple of months. Then, after a couple of months of "time off," move it closer to the light again and increase the fertilizer to what you were originally giving it. After a few weeks of increased light and fertilizer, the orchid should initiate a brand new spike from the base of the plant. You may get one or three or more...it all depends on the plant. But either way it's tremendously gratifying to see a new spike shoot out from the base of the plant. Then you get to start the process all over again!
As far as whether your orchid is healthy enough to give another flowering...it sounds like it is, but in my opinion, it's always best to let the orchid decide what it wants to do next. It's my thinking that if you snip the spike close to the base, you're pretty much forcing it into flowering a certain way... whereas if you cut it just below the first bud, you are only removing the spent portion of the stalk and you are allowing the orchid to decide what it wants to do next. Then you don't even need to worry about whether the orchid looks healthy enough for another flowering; the orchid will decide if it wants to flower or if it wants to take a break. I personally feel that you create the least amount of stress and injury by just removing the part the plant has used up (the very top of the stalk). Of course, this is just my opinion, and I know some people do it differently...but I've never gone wrong using this method.
Anyway, I hope this rather long-winded response was helpful! I'd love to see a pic of your pretty phal before it drops its flowers! Post one if you get a chance...although I think we might be hijacking Sand Tiger's thread here...you could always message me with a pic too, or if you have any more questions. I'd love for you to keep me posted on how your plant is doing!
Best of luck!