"Normal" building movement would only be due to wind or earthquake. Movement due to normal wind would only be slightly apparent to very sensitive individuals in very tall buildings, as in dozens of floors. Most people would notice slight wind movement in very high winds or minor earthquakes, but nothing alarming. The only alarming movement anyone would notice would be during strong earthquakes.
How much a building moves due to a lateral force such as wind or earthquake depends partly on the buildings height to width ratio and the relative rigidity of the building's structural elements. The critical width when determining the height to width ration is the building's critical (narrowest) depth in the direction of the force. Since wind and earthquakes can come from any direction, we can simplify and say the least horizontal dimension.
Sometimes people in buildings will sense vibrations caused by passing trains, or even heavily loaded trucks. Some types of soils have a certain elasticity that transmits vibrations from heavy movement into the building's foundation and up into the structure. These movements are primarily vertical in direction and would typically not be threatening to the building, though repeated severe vibrations can cause brittle finish materials such as plaster work to crack or even fall out of position.
A similar movement can be caused just from people moving normally on a particular floor system, especially in smaller buildings, because the floor structure is overly elastic. This usually would not indicate any structural failure is imminent, rather that the structure was not built adequately from a usability standpoint.
Renovations to existing buildings must be carefully planned to avoid disturbing structural, load bearing elements. Not just bearing vertical loads, but lateral loads as well. If they must be disturbed, adequate temporary shoring must be in place until proper permanent structure is replaced. Even if shoring was not done, building movement would not be discernible unless structural failure was imminent. Imminent in building failure can be anywhere from within the next few minutes to happening over a period of several months.
Besides wind, earthquake, heavy vehicles, and compromised structure, the only other cause for building movement would be differential settlement due to foundation not being founded in stable soil materials. This is actually a subset of structural failure, as stable soils is a necessary part of a sound structure.
When an engineer assesses a structure for soundness, testing may or may not be done, depending on the situation. Structural testing often means testing to failure, which is often not feasible in an occupied structure. The engineer will make an assessment based mainly on inspection of existing conditions and comparing with the building's plans to see if it was built in accordance to the plans, if the plans have design errors, if there are signs of failure such as cracks of certain form in certain locations or sagging and deformation of structural elements.
Some small scale materials tests can be done to verify the installed material matches the design specifications by taking samples and testing in a laboratory. Only after considering all available evidence will the engineer make an assessment based on the evidence and his/her experience and knowledge. They would typically publish a report summarizing what they did, what they observed and their findings. While such reports are quite technical by nature, the main gist of it should be comprehensible to any reasonably intelligent person without an engineering background.
It's impossible to say if you overreacted since I was not there. While there is certainly cause for concern, imminent collapse World Trade Center style is highly unusual. I've seen many obviously failed buildings in developing countries in which squatters live without incident for years. Yet one would have to be truly desperate to spend time in such buildings, as sudden catastrophic collapse does remain a possibility.
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The joists are undersize or the center distance is too great to provide the stiffness you want.It is likely strong enough unless you want large aquariums.