Suppose there is a 10x10 grid.
Can you tile it completely with 1x4 tiles (oriented in any direction)?
No, it is impossible.
Proof
Label the grid cells with letters A-D as follows, filling in each diagonal line with each successive label:
Note that anywhere you place a 1x4 tile, it will cover one of each label A-D. Thus each letter will be covered the same number of times.
However, the grid has an unequal count of each letter. Note that each row has either two or three of each letter. Looking at just the letters A and B:
There are 5 rows with 3 As, but 6 rows with 3 Bs. Hence there is one more B than As in the grid. Thus any tiling that covers every A will leave a B uncovered.