Translating backup size to database size
You could have asked mysql for all individual table sizes

SELECT table_schema,table_name,data_length,index_length
FROM information_schema.tables
WHERE table_schema NOT IN ('information_schema','mysql');


Summary by Storage Engine

SELECT IFNULL(B.engine,'Total') "Storage Engine",
CONCAT(LPAD(REPLACE(FORMAT(B.DSize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Data Size",
CONCAT(LPAD(REPLACE(FORMAT(B.ISize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Index Size",
CONCAT(LPAD(REPLACE(FORMAT(B.TSize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Table Size" FROM
(SELECT engine,SUM(data_length) DSize,SUM(index_length) ISize,
SUM(data_length+index_length) TSize FROM information_schema.tables
WHERE table_schema NOT IN ('mysql','information_schema','performance_schema')
AND engine IS NOT NULL GROUP BY engine WITH ROLLUP) B,(SELECT 3 pw) A
ORDER BY TSize;


Summary by Database

SELECT DBName,CONCAT(LPAD(FORMAT(SDSize/POWER(1024,pw),3),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Data Size",
CONCAT(LPAD(FORMAT(SXSize/POWER(1024,pw),3),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Index Size",
CONCAT(LPAD(FORMAT(STSize/POWER(1024,pw),3),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Total Size"
FROM (SELECT IFNULL(DB,'All Databases') DBName,SUM(DSize) SDSize,
SUM(XSize) SXSize,SUM(TSize) STSize FROM (SELECT table_schema DB,
data_length DSize,index_length XSize,data_length+index_length TSize
FROM information_schema.tables WHERE table_schema NOT IN
('mysql','information_schema','performance_schema')) AAA
GROUP BY DB WITH ROLLUP) AA,(SELECT 3 pw) BB ORDER BY (SDSize+SXSize);


Summary by Database/Storage Engine

SELECT Statistic,DataSize "Data Size",IndexSize "Index Size",TableSize "Table Size"
FROM (SELECT IF(ISNULL(table_schema)=1,10,0) schema_score,
IF(ISNULL(engine)=1,10,0) engine_score, IF(ISNULL(table_schema)=1,
'ZZZZZZZZZZZZZZZZ',table_schema) schemaname,
IF(ISNULL(B.table_schema)+ISNULL(B.engine)=2,
"Storage for All Databases",IF(ISNULL(B.table_schema)+ISNULL(B.engine)=1,
CONCAT("Storage for ",B.table_schema),CONCAT(B.engine," Tables for ",
B.table_schema))) Statistic,
CONCAT(LPAD(REPLACE(FORMAT(B.DSize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') DataSize,
CONCAT(LPAD(REPLACE(FORMAT(B.ISize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') IndexSize,
CONCAT(LPAD(REPLACE(FORMAT(B.TSize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') TableSize FROM (SELECT table_schema,engine,
SUM(data_length) DSize,SUM(index_length) ISize,SUM(data_length+index_length) TSize
FROM information_schema.tables WHERE table_schema NOT IN
('mysql','information_schema','performance_schema') AND engine IS NOT NULL
GROUP BY table_schema,engine WITH ROLLUP) B,(SELECT 3 pw) A) AA
ORDER BY schemaname,schema_score,engine_score;


Please send these queries to your IT people and have them email the results back

Please keep in mind that mysqldumps do not contain indexes. A mysqldump is simply a logical representation of the data plus the commands and directive to crate the table, load the table, and make indexes. What get generated physically upon restore can be known by these queries before launching the mysqldump.