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That's not correct...at least not on the -700. If you're too heavy for the altitude you pop in (like FL400 or 410), it'll give that message.
That is correct, with a caveat, you must enter the actual temperature (ISA+) in the performance page in order for the FMS to calculate the max altitude properly. At my company we aren't supposed to use the FMS performance data anyway, but it's good to know the airplane thinks it will make it.
GCD
Have regularly climbed the -200 at 250/.7?
I have as that's my companies profile. I find I have the ability to climb to filed altitudes with a comfortable buffer. Perhaps my point of contention is with the term of higher flight level. If the -200 won't make it to a particularly high altitude at one speed, it likely won't or shouldn't be at attempted at any other. When however it's operated within it's proper (not it's certified) envelope, 250 to .7 provides a much better climb rate than 290 to .7 and provides more than an adequate level of safety.
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290kias until FL200 - then VS 500ft/min the rest of the way up.