This issue has only been raised in the community once, two years ago, to the best of my knowledge. I think it's high time to raise the issue again!
readOGR, from the rgdal package, cannot handle multipoint shapefiles, even though they load fine in other GIS packages like QGIS. Of course, you can save the file as a single part object (see image below) in an external program, but ideally it would also be possible to do it in R. Please see worked example below.

download.file("http://www.personal.leeds.ac.uk/~georl/egs/lnd-stations-multi.zip",
"lnd-stations.zip") # download multi part polygon
unzip("lnd-stations.zip") # unzip
lndS <- readOGR(".", "lnd-stations", p4s = "+init=epsg:27700") # load
OGR data source with driver: ESRI Shapefile
Source: ".", layer: "lnd-stations"
with 2532 features and 27 fields
Feature type: wkbMultiPoint with 2 dimensions
Error in readOGR(".", "lnd-stations", p4s = "+init=epsg:27700") :
Incompatible geometry: 4
After performing "Multipart to singleparts" in the QGIS Vector menu (illustrated above) and saving the file, it loads fine in R. Try it:
download.file("http://www.personal.leeds.ac.uk/~georl/egs/lnd-stns.zip",
"lnd-stns.zip")
unzip("lnd-stns.zip")
lndS <- readOGR(".", "lnd-stns", p4s = "+init=epsg:27700")
plot(lndS)