This has been an issue with the MODIS Reprojection Tool, as the files you are trying to download are placed in different folders on their server and the code often fails to parse through each of them. I have a small script that essentially does what the MRT does.
Use the MODIS Data Download Manager through the LP DAAC web portal.
1) This link will take you to the portal with the download manager. Click on LPDAAC2Disk Download Manager. Here, you have two options - One, to download the manager or use the web interface.
2) I suggest clicking on the web interface. That will take you to this page.
3) Now draw a polygon around your region. Click on the dataset you need, the time period required and click submit. The download usually takes around 2-3 hours depending on the number of files and the size of the region selected.
4) One can open these files in ArcMap and use the Extract SubDataset tool to get either EVI or NDVI or the data that you need. Alternately, store all these files in a folder and use the python script below to perform the same.
arcpy.env.workspace = "F:\\MODIS Data\\"
out_dir = "F:\\MODIS Data\\EVIData\\"
rasters = arcpy.ListRasters("*")
count = 1
for raster in rasters:
output = str(raster)
s1 = raster.split('.')
name = out_dir + raster.split('.') + s1
arcpy.ExtractSubDataset_management(raster, name, "1")
5) Note, you are basically manually doing what the reprojection tool does. So you are extracting a subdataset and then you need to project it to the projection you need.