Hutchinson, B. (2002), an article in the book Synergy Matters by Adrian M. Castell may help in understanding what is meant by bottom up thinking and the opposite, top down thinking.
[T]he term ‘top down’ is used to describe an approach to problem solving where the problem space is defined first. The worldviews of the participants are used to conceptualise the desired state of the proposed system. Once this is achieved, the system is developed within this boundary. The components, or subsystems are derived within the context of the predefined desired state. The term ‘bottom up’ is used to describe an approach where no assumptions are made about the boundary of the problem space. The behaviour of component parts are rationally observed to determine the properties they have. Management decisions, or system designs are then based on the observed behaviour of components.
An example given in the conclusions is:
[I]nvestigating a system failure in bottom up mode would look for causes at the element level, or more precisely, at their interactions. The top down mode would tend to look at the overall system, and would put ‘blame’ on the system itself and how it functions, rather than emergent properties of system element interactions.
Think of it like a brick building. You need to build it from the bottom up, with each brick being correctly placed to form the correct shape etc. If the bricks are placed differently, you may get a different building, or maybe you will get the same building but badly constructed.
Take the following as a more detailed example comparison...
There is a team of researchers working on a task and each member inputs their findings into a computer system which analyses the results and comes up with a conclusion.
The program relies on accurate information from each team member entry but there was a slight error in a couple of entries producing an incorrect conclusion by the system.
A top down thinker would blame the computer system analysing the results. A bottom up thinker would analyse what the system was analysing and how, leading to them finding the real cause.
Hutchinson, B. (2002). Bottom up thinking. In Synergy Matters (pp. 445-450). Springer, Boston, MA. doi: 10.1007/0-306-47467-0_75
Preview available in Google Books