Based on an old comment from NotEvans, it turns out that the reported NMR data is from the end-of-chapter problems of Organic Chemistry 2ed, Clayden et al. (Problem 11, Chapter 18). These can be accessed online using the username and password provided in the front cover of the textbook.
δH (ppm) 0.89 (6H, s), 1.35-1.70 (1H, broad m), 1.41* (1H, s), 1.96 (2H, s), 2.06 (2H, t, J 6 Hz), 4.11 (2H, d, J 7 Hz), and 5.48 (1H, t, J 7 Hz).
There is an erratum in this data; the "broad multiplet" should integrate to four protons, not one. This is not officially corrected in the published list of errata, but is sufficiently obvious that I am willing to go on record to say it is a typo.
In my experience, allylic couplings can sometimes be difficult to resolve, unless you manipulate the spectrum (e.g. with a suitable apodisation / window function). So, the lack of allylic coupling in the data shown above is not entirely surprising.
With this knowledge in hand, the spectral data can be reasonably easily assigned, and is handled admirably by Jan's answer.
However, there is something that I wanted to comment on, and it is that counter to intuition, small couplings (e.g. allylic coupling, ~1.5 to 2 Hz) are typically easier to see with lower field strength, not higher. It's because if you want to cover a spectral width of 20 ppm at 600 MHz, that corresponds to 1200 Hz (vs 400 Hz at 200 MHz). If all the experiment parameters are kept the same, the digital resolution (in terms of Hz per point) at 600 MHz is 3 times poorer than at 200 MHz.
This might not be the most instructive example, but I am not particularly bothered to find a better one. This is a methyl group which is displaying one large coupling ($^3\!J$) and two small couplings (one $^4\!J$ and one $^5\!J$). You can see that the resolution is ever-so-slightly better in the 400 MHz spectrum. Both spectra were acquired with the same spectral width (20 ppm), the same number of points in the FID (64k), and no zero filling has been applied. As a result, the digital resolution in the 500 MHz spectrum is 1.25 times worse (0.32 Hz vs 0.24 Hz for the 400 MHz).